liberal arts math assignment need answer today
Thinking
Mathematically Seventh Edition
Robert Blitzer Miami Dade College
Director, Portfolio Management Anne Kelly
Courseware Portfolio Managers Marnie Greenhut and Dawn Murrin
Courseware Portfolio Management Assistant Stacey Miller
Content Producer Kathleen A. Manley
Managing Producer Karen Wernholm
Producer Nick Sweeny
Manager, Courseware QA Mary Durnwald
Product Marketing Manager Kyle DiGiannantonio
Field Marketing Manager Andrew Noble
Marketing Assistant Brooke Imbornone
Senior Author Support/Technology Specialist Joe Vetere
Manager, Rights and Permissions Gina Cheselka
Manufacturing Buyer Carol Melville, LSC Communications
Text and Cover Design Studio Montage
Production Coordination and Composition codeMantra
Illustrations Scientific Illustrations
Cover Images Catherine Ledner/Iconica/Getty Images (cow)
and Hunter Bliss/Shutterstock (frame)
Copyright © 2019, 2015, 2011 by Pearson Education, Inc. All Rights Reserved. Printed in the United States of America.
This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited
reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate
contacts within the Pearson Education Global Rights & Permissions department, please visit
www.pearsoned.com/permissions/.
Attributions of third party content appear on page C1, which constitutes an extension of this copyright page.
PEARSON, ALWAYS LEARNING, and MYLAB are exclusive trademarks owned by Pearson Education, Inc. or its
affiliates in the U.S. and/or other countries.
Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their
respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or
descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or
promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson
Education, Inc. or its affiliates, authors, licensees or distributors.
Library of Congress Cataloging-in-Publication Data Names: Blitzer, Robert, author.
Title: Thinking mathematically / Robert F. Blitzer.
Description: Seventh edition. | Boston : Pearson, [2019]
Identifiers: LCCN 2017046337 | ISBN 9780134683713 (alk. paper) | ISBN 0134683714 (alk. paper)
Subjects: LCSH: Mathematics–Textbooks.
Classification: LCC QA39.3 .B59 2019 | DDC 510–dc23
LC record available at https://lccn.loc.gov/2017046337
ISBN-13: 978-0-13-468371-3
ISBN-10: 0-13-468371-4
Contents About the Author vi
Preface vii
Resources for Success ix
To the Student xi
Acknowledgments xii
Index of Applications xv
1 Problem Solving and Critical Thinking 1
1.1 Inductive and Deductive
Reasoning 2
1.2 Estimation, Graphs,
and Mathematical
Models 14
1.3 Problem Solving 30
Chapter Summary,
Review, and Test 43
Chapter 1 Test 46
2 Set Theory
49
2.1 Basic Set Concepts 50
2.2 Subsets 64
2.3 Venn Diagrams and
Set Operations 73
2.4 Set Operations and
Venn Diagrams with
Three Sets 87
2.5 Survey Problems 99
Chapter Summary,
Review, and Test 110
Chapter 2 Test 114
3 Logic 117
3.1 Statements, Negations,
and Quantified Statements 118
3.2 Compound Statements
and Connectives 126
3.3 Truth Tables for
Negation, Conjunction,
and Disjunction 139
3.4 Truth Tables for the Conditional
and the Biconditional 154
3.5 Equivalent Statements and Variations
of Conditional Statements 166
3.6 Negations of Conditional Statements
and De Morgan’s Laws 176
3.7 Arguments and Truth Tables 184
3.8 Arguments and Euler Diagrams 199
Chapter Summary, Review, and Test 209
Chapter 3 Test 213
4 Number Representation
and Calculation 215
4.1 Our Hindu-Arabic
System and Early
Positional Systems 216
4.2 Number Bases in
Positional Systems 224
4.3 Computation in
Positional Systems 231
4.4 Looking Back at Early
Numeration Systems 240
Chapter Summary,
Review, and Test 247
Chapter 4 Test 250
iii
iv Contents
5 Number Theory and
the Real Number
System
251
5.1 Number Theory:
Prime and Composite
Numbers 252
5.2 The Integers; Order
of Operations 262
5.3 The Rational
Numbers 276
5.4 The Irrational
Numbers 291
5.5 Real Numbers and
Their Properties;
Clock Addition 304
5.6 Exponents and Scientific
Notation 315
5.7 Arithmetic and Geometric
Sequences 326
Chapter Summary, Review, and Test 336
Chapter 5 Test 341
6 Algebra:
Equations and
Inequalities 343
6.1 Algebraic
Expressions
and Formulas 344
6.2 Linear Equations
in One Variable and Proportions 354
6.3 Applications of Linear Equations 369
6.4 Linear Inequalities in One Variable 380
6.5 Quadratic Equations 390
Chapter Summary, Review, and Test 405
Chapter 6 Test 409
7 Algebra: Graphs,
Functions,
and Linear
Systems 411
7.1 Graphing and
Functions 412
7.2 Linear Functions
and Their Graphs 424
7.3 Systems of Linear
Equations in
Two Variables 438
7.4 Linear Inequalities
in Two Variables 453
7.5 Linear Programming 462
7.6 Modeling Data: Exponential,
Logarithmic, and Quadratic Functions 468
Chapter Summary, Review, and Test 484
Chapter 7 Test 490
8 Personal
Finance
493
8.1 Percent,
Sales Tax,
and Discounts 494
8.2 Income Tax 503
8.3 Simple
Interest 514
8.4 Compound Interest 519
8.5 Annuities, Methods of Saving,
and Investments 529
8.6 Cars 545
8.7 The Cost of Home Ownership 554
8.8 Credit Cards 563
Chapter Summary, Review, and Test 572
Chapter 8 Test 578
Contents v
9 Measurement
581
9.1 Measuring Length;
The Metric System 582
9.2 Measuring Area
and Volume 592
9.3 Measuring Weight
and Temperature 602
Chapter Summary, Review, and Test 611
Chapter 9 Test 614
10 Geometry
615
10.1 Points, Lines,
Planes, and Angles 616
10.2 Triangles 625
10.3 Polygons, Perimeter, and Tessellations 637
10.4 Area and Circumference 646
10.5 Volume and Surface Area 657
10.6 Right Triangle Trigonometry 666
10.7 Beyond Euclidean Geometry 676
Chapter Summary, Review, and Test 685
Chapter 10 Test 691
11
Counting Methods
and Probability
Theory
693
11.1 The Fundamental
Counting Principle 694
11.2 Permutations 700
11.3 Combinations 708
11.4 Fundamentals of Probability 715
11.5 Probability with the Fundamental
Counting Principle, Permutations,
and Combinations 724
11.6 Events Involving Not and Or; Odds 731
11.7 Events Involving And; Conditional
Probability 744
11.8 Expected Value 756
Chapter Summary, Review, and Test 763
Chapter 11 Test 769
12 Statistics
771
12.1 Sampling, Frequency
Distributions, and
Graphs 772
12.2 Measures of
Central Tendency 786
12.3 Measures of Dispersion 800
12.4 The Normal Distribution 808
12.5 Problem Solving with the Normal
Distribution 822
12.6 Scatter Plots, Correlation, and
Regression Lines 827
Chapter Summary, Review, and Test 838
Chapter 12 Test 843
13 Voting and
Apportionment 845
13.1 Voting Methods 846
13.2 Flaws of Voting
Methods 858
13.3 Apportionment
Methods 869
13.4 Flaws of
Apportionment
Methods 883
Chapter Summary, Review, and Test 893
Chapter 13 Test 896
14 Graph Theory 897
14.1 Graphs, Paths,
and Circuits 898
14.2 Euler Paths and Euler Circuits 908
14.3 Hamilton Paths and Hamilton Circuits 920
14.4 Trees 930
Chapter Summary, Review, and Test 939
Chapter 14 Test 944
Answers to Selected Exercises AA1
Subject Index I1
Credits C1
About the Author
Bob Blitzer is a native of Manhattan and received a Bachelor of Arts degree with dual majors in mathematics and psychology (minor: English literature) from
the City College of New York. His unusual combination of academic interests
led him toward a Master of Arts in mathematics from the University of Miami
and a doctorate in behavioral sciences from Nova University. Bob’s love for
teaching mathematics was nourished for nearly 30 years at Miami Dade College,
where he received numerous teaching awards, including Innovator of the Year
from the League for Innovations
in the Community College and an
endowed chair based on excellence in
the classroom. In addition to Thinking Mathematically, Bob has written textbooks covering introductory
algebra, intermediate algebra, college
algebra, algebra and trigonometry,
precalculus, trigonometry, and liberal
arts mathematics for high school
students, all published by Pearson.
When not secluded in his Northern
California writer’s cabin, Bob can be
found hiking the beaches and trails
of Point Reyes National Seashore,
and tending to the chores required
by his beloved entourage of horses,
chickens, and irritable roosters.
vi
Preface
Thinking Mathematically, Seventh Edition provides a general survey of mathematical topics that are useful
in our contemporary world. My primary purpose in
writing the book was to show students how mathematics
can be applied to their lives in interesting, enjoyable,
and meaningful ways. The book’s variety of topics and
flexibility of sequence make it appropriate for a one- or
two-term course in liberal arts mathematics, quantitative
reasoning, finite mathematics, as well as for courses
specifically designed to meet state-mandated requirements
in mathematics.
I wrote the book to help diverse students, with
different backgrounds and career plans, to succeed.
Thinking Mathematically, Seventh Edition, has four major goals:
1. To help students acquire knowledge of fundamental mathematics.
2. To show students how mathematics can solve authentic problems that apply to their lives.
3. To enable students to understand and reason with quantitative issues and mathematical ideas they are
likely to encounter in college, career, and life.
4. To enable students to develop problem-solving skills, while fostering critical thinking, within an interesting
setting.
One major obstacle in the way of achieving these
goals is the fact that very few students actually read their
textbook. This has been a regular source of frustration
for me and my colleagues in the classroom. Anecdotal
evidence gathered over years highlights two basic reasons
why students do not take advantage of their textbook:
“I’ll never use this information.”
“I can’t follow the explanations.”
I’ve written every page of the Seventh Edition with the
intent of eliminating these two objections. The ideas and
tools I’ve used to do so are described for the student in
“A Brief Guide to Getting the Most from This Book,”
which appears inside the front cover.
What’s New in the Seventh Edition?
• New and Updated Applications and Real-World Data. I’m on a constant search for real-world data that can
be used to illustrate unique mathematical applications.
I researched hundreds of books, magazines,
newspapers, almanacs, and online sites to prepare the
Seventh Edition. This edition contains 110 worked-out
examples and exercises based on new data sets and
104 examples and exercises based on updated data.
New applications include student-loan debt (Exercise
Set 1.2), movie rental options (Exercise Set 1.3),
impediments to academic performance (Section 2.1),
measuring racial prejudice, by age (Exercise Set 2.1),
generational support for legalized adult marijuana
use (Exercise Set 2.3), different cultural values among
nations (Exercise Set 2.5), episodes from the television
series The Twilight Zone (Section 3.6) and the film Midnight Express (Exercise Set 3.7), excuses by college students for not meeting assignment deadlines
(Exercise Set 5.3), fraction of jobs requiring various
levels of education by 2020 (Exercise Set 5.3), average
earnings by college major (Exercise Set 6.5), the pay gap
(Exercise Set 7.2), inmates in federal prisons for drug
offenses and all other crimes (Exercise Set 7.3), time
breakdown for an average 90-minute NFL broadcast
(Section 11.6), Scrabble tiles (Exercise Set 11.5), and
are inventors born or made? (Section 12.2).
• New Blitzer Bonuses. The Seventh Edition contains a variety of new but optional enrichment essays. There
are more new Blitzer Bonuses in this edition than in any
previous revision of Thinking Mathematically. These include “Surprising Friends with Induction” (Section 1.1),
“Predicting Your Own Life Expectancy” (Section 1.2),
“Is College Worthwhile?” (Section 1.2), “Yogi-isms”
(Section 3.4), “Quantum Computers” (Section 4.3),
“Slope and Applauding Together” (Section 7.2),
“A Brief History of U.S. Income Tax” (Section 8.2)
“Three Decades of Mortgages” (Section 8.7), “Up to
Our Ears in Debt” (Section 8.8), “The Best Financial
Advice for College Graduates” (Section 8.8), “Three
Weird Units of Measure” (Section 9.1), “Screen Math”
(Section 10.2), “Senate Voting Power” (Section 13.3),
“Hamilton Mania” (Section 13.3), “Dirty Presidential
Elections” (Section 13.3), “Campaign Posters as Art”
(Section 13.4), and “The 2016 Presidential Election”
(Section 13.4).
• New Graphing Calculator Screens. All screens have been updated using the TI-84 Plus C.
• Updated Tax Tables. Section 8.2 (Income Tax) contains the most current federal marginal tax tables
and FICA tax rates available for the Seventh Edition.
• New MyLabTM Math. In addition to the new functionalities within an updated MyLab Math,
the new items specific to Thinking Mathematically, Seventh Edition MyLab Math include
~ All new objective-level videos with assessment
~ Interactive concept videos with assessment
~ Animations with assessment
~ StatCrunch integration.
vii
viii Preface
What Familiar Features Have Been Retained in the
Seventh Edition?
• Chapter-Opening and Section-Opening Scenarios. Every chapter and every section open with a scenario
presenting a unique application of mathematics in
students’ lives outside the classroom. These scenarios
are revisited in the course of the chapter or section
in an example, discussion, or exercise. The often
humorous tone of these openers is intended to help
fearful and reluctant students overcome their negative
perceptions about math. A feature called “Here’s
Where You’ll Find These Applications” is included
with each chapter opener.
• Section Objectives (What Am I Supposed to Learn?). Learning objectives are clearly stated at the beginning
of each section. These objectives help students
recognize and focus on the section’s most important
ideas. The objectives are restated in the margin at their
point of use.
• Detailed Worked-Out Examples. Each example is titled, making the purpose of the example clear.
Examples are clearly written and provide students with
detailed step-by-step solutions. No steps are omitted
and each step is thoroughly explained to the right of
the mathematics.
• Explanatory Voice Balloons. Voice balloons are used in a variety of ways to demystify mathematics.
They translate mathematical language into everyday
English, help clarify problem-solving procedures,
present alternative ways of understanding concepts,
and connect problem solving to concepts students
have already learned.
• Check Point Examples. Each example is followed by a similar matched problem, called a Check Point,
offering students the opportunity to test for conceptual
understanding by working a similar exercise. The
answers to the Check Points are provided in the answer
section in the back of the book. Worked-out video
solutions for many Check Points are in the MyLab
Math course.
• Great Question! This feature presents study tips in the context of students’ questions. Answers to the
questions offer suggestions for problem solving, point
out common errors to avoid, and provide informal
hints and suggestions. As a secondary benefit, this
feature should help students not to feel anxious or
threatened when asking questions in class.
• Brief Reviews. The book’s Brief Review boxes summarize mathematical skills that students should
have learned previously, but which many students
still need to review. This feature appears whenever a
particular skill is first needed and eliminates the need
to reteach that skill.
• Concept and Vocabulary Checks. The Seventh Edition contains 653 short-answer exercises, mainly fill-in-
the blank and true/false items, that assess students’
understanding of the definitions and concepts
presented in each section. The Concept and Vocabulary
Checks appear as separate features preceding the
Exercise Sets. These are assignable in the MyLab Math
course.
• Extensive and Varied Exercise Sets. An abundant collection of exercises is included in an Exercise Set at
the end of each section. Exercises are organized within
seven category types: Practice Exercises, Practice
Plus Exercises, Application Exercises, Explaining the
Concepts, Critical Thinking Exercises, Technology
Exercises, and Group Exercises.
• Practice Plus Problems. This category of exercises contains practice problems that often require students
to combine several skills or concepts, providing
instructors the option of creating assignments that
take Practice Exercises to a more challenging level.
• Chapter Summaries. Each chapter contains a review chart that summarizes the definitions and concepts in
every section of the chapter. Examples that illustrate
these key concepts are also referenced in the chart.
• End-of-Chapter Materials. A comprehensive collection of review exercises for each of the chapter’s sections
follows the Summary. This is followed by a Chapter
Test that enables students to test their understanding of
the material covered in the chapter. Worked-out video
solutions are available for every Chapter Test Prep
problem in the MyLab Math course or on YouTube.
• Learning Guide. This study aid is organized by objective and provides support for note-taking,
practice, and video review. The Learning Guide
is available as PDFs in MyLab Math. It can also
be packaged with the textbook and MyLab Math
access code.
I hope that my love for learning, as well as my respect
for the diversity of students I have taught and learned from
over the years, is apparent throughout this new edition.
By connecting mathematics to the whole spectrum of
learning, it is my intent to show students that their world is
profoundly mathematical, and indeed, p is in the sky.
Robert Blitzer
pearson.com/mylab/math
Resources for Success MyLab
TM Math Online Course for
Thinking Mathematically, Seventh Edition
by Robert Blitzer (access code required) MyLab Math is available to accompany Pearson’s market leading text offerings. To give
students a consistent tone, voice, and teaching method each text’s flavor and approach
are tightly integrated throughout the accompanying MyLab Math course, making
learning the material as seamless as possible.
NEW! Video Program
All new objective-level videos provide
a new level of coverage throughout the
text. Videos at the objective level allow
students to get support just where they
need it. Instructors can assign these as
media assignments or use the provided
assessment questions for each video.
NEW! Interactive
Concept Videos
New Interactive Concept Videos are also
available in MyLab Math. After a brief
explanation, the video pauses to ask
students to try a problem on their own.
Incorrect answers are followed by further
explanation, taking into consideration what
may have led to the student selecting
that particular wrong answer. Incorrect
answer ‘A’ goes down one path while
incorrect answer ‘B’ provides a different
explanation based on why the student may
have selected that option.
NEW! Animations
New animations let students interact with
the math in a visual, tangible way. These
animations allow students to explore and
manipulate the mathematical concepts,
leading to more durable understanding.
Corresponding exercises in MyLab Math
make these truly assignable.
StatCrunch Newly integrated StatCrunch allows
students to harness technology to
perform complex analyses on data.
Resources for Success ix
Instructor Resources
Annotated Instructor’s Edition (AIE) ISBN-10: 0-13-468454-0
ISBN-13: 978-0-13-468454-3
The AIE includes answers to all exercises presented in
the book, most on the page with the exercise and the
remainder in the back of the book.
The following resources can be downloaded from
MyLab Math or the Instructor’s Resource Center on
www.pearsonhighered.com.
MyLab Math with Integrated Review Provides a full suite of supporting resources for the
collegiate course content plus additional assignments
and study aids for students who will benefit from
remediation. Assignments for the integrated review
content are preassigned in MyLab™ Math, making it
easier than ever to create your course.
Instructor’s Solutions Manual This manual contains detailed, worked-out solutions to
all the exercises in the text.
PowerPoint Lecture Presentation These editable slides present key concepts and
definitions from the text. Instructors can add art from
the text located in the Image Resource Library in MyLab
Math or slides that they create on their own. PointPoint
slides are fully accessible.
Image Resource Library This resource in MyLab Math contains all art from the
text, for instructors to use in their own presentations
and handouts.
Instructor’s Testing Manual The Testing Manual includes two alternative tests per
chapter. These items may be used as actual tests or as
references for creating actual tests.
TestGen TestGen® (www.pearsoned.com/testgen) enables
instructors to build, edit, print, and administer tests
using a computerized bank of questions developed
to cover all the objectives of the text. TestGen is
algorithmically based, allowing instructors to create
multiple but equivalent versions of the same question
or test with the click of a button. Instructors can also
modify test bank questions or add new questions. The
software are available for download from Pearson’s
Instructor Resource Center.
Student Resources Learning Guide with Integrated Review Worksheets
ISBN 10: 0-13-470508-4
ISBN 13: 978-0-13470508-8
Bonnie Rosenblatt, Reading Area Community College
This workbook is organized by objective and provides
support for note-taking, practice, and video review and
includes the Integrated Review worksheets from the
Integrated Review version of the MyLab Math course.
The Learning Guide is also available as PDFs in MyLab
Math. It can also be packaged with the textbook and
MyLab Math access code.
Student’s Solutions Manual ISBN 10: 0-13-468650-0
ISBN 13: 978-0-13-468650-9
Daniel Miller, Niagara County Community College
This manual provides detailed, worked-out solutions
to odd-numbered exercises, as well as solutions to all
Check Points, Concept and Vocabulary Checks, Chapter
Reviews, and Chapter Tests.
pearson.com/mylab/math
Resources for Success
x Resources for Success
To the Student The bar graph shows some of the qualities that students say make a great teacher.
It was my goal to incorporate each of these qualities throughout the pages of this
book to help you gain control over the part of your life that involves numbers and
mathematical ideas.
Explains Things Clearly
I understand that your primary purpose in reading Thinking Mathematically is to acquire a solid understanding of the required topics in your
liberal arts math course. In order to achieve this goal, I’ve carefully
explained each topic. Important definitions and procedures are
set off in boxes, and worked-out examples that present solutions
in a step-by-step manner appear in every section. Each example is
followed by a similar matched problem, called a Check Point, for you
to try so that you can actively participate in the learning process as
you read the book. (Answers to all Check Points appear in the back
of the book and video solutions are in MyLab Math.)
Funny & Entertaining
Who says that a math textbook can’t be entertaining? From our
engaging cover to the photos in the chapter and section openers, prepare
to expect the unexpected. I hope some of the book’s enrichment essays,
called Blitzer Bonuses, will put a smile on your face from time to time.
Helpful
I designed the book’s features to help you acquire knowledge of
fundamental mathematics, as well as to show you how math can solve authentic
problems that apply to your life. These helpful features include
• Explanatory Voice Balloons: Voice balloons are used in a variety of ways to make math less intimidating. They translate mathematical language into everyday English,
help clarify problem-solving procedures, present alternative ways of understanding
concepts, and connect new concepts to concepts you have already learned.
• Great Question!: The book’s Great Question! boxes are based on questions students ask in class. The answers to these questions give suggestions for problem solving,
point out common errors to avoid, and provide informal hints and suggestions.
• Chapter Summaries: Each chapter contains a review chart that summarizes the definitions and concepts in every section of the chapter. Examples from the
chapter that illustrate these key concepts are also referenced in the chart. Review
these summaries and you’ll know the most important material in the chapter!
Passionate about the Subject
I passionately believe that no other discipline comes close to math in offering a more
extensive set of tools for application and development of your mind. I wrote the book
in Point Reyes National Seashore, 40 miles north of San Francisco. The park consists
of 75,000 acres with miles of pristine surf-washed beaches, forested ridges, and bays
bordered by white cliffs. It was my hope to convey the beauty and excitement of
mathematics using nature’s unspoiled beauty as a source of inspiration and creativity.
Enjoy the pages that follow as you empower yourself with the mathematics needed
to succeed in college, your career, and in your life.
Regards,
Bob Robert Blitzer
xi
An enormous benefit of authoring a successful textbook
is the broad-based feedback I receive from students,
dedicated users, and reviewers. Every change to this
edition is the result of their thoughtful comments and
suggestions. I would like to express my appreciation to all
the reviewers, whose collective insights form the backbone
of this revision. In particular, I would like to thank the
following people for reviewing Thinking Mathematically for this Seventh Edition.
Deana Alexander, Indiana University—Purdue University Nina Bohrod, Anoka-Ramsey Community College Kim Caldwell, Volunteer State Community College Kevin Charlwood, Washburn University Elizabeth T. Dameron, Tallahassee Community College Darlene O. Diaz, Santiago Canyon College Cornell Grant, Georgia Piedmont Technical College Theresa Jones, Texas State University Elizabeth Kiedaisch, College of DuPage Lauren Kieschnick, Mineral Area College Alina Klein, University of Dubuque Susan Knights, College of Western Idaho Isabelle Kumar, Miami Dade College Dennine LaRue, Farmont State University David Miller, William Paterson University Carla A. Monticelli, Camden County College Tonny Sangutei, North Carolina Central University Cindy Vanderlaan, Indiana Purdue University —Fort Wayne Alexandra Verkhovtseva, Anoka-Ramsey Community College
Each reviewer from every edition has contributed to
the success of this book and I would like to also continue
to offer my thanks to them.
David Allen, Iona College; Carl P. Anthony, Holy Family University; Laurel Berry, Bryant and Stratton College; Kris Bowers, Florida State University; Gerard Buskes, University of Mississippi; Fred Butler, West Virginia University; Jimmy Chang, St. Petersburg College; Jerry Chen, Suffolk County Community College; Ivette Chuca, El Paso Community College; David Cochener, Austin Peay State University; Stephanie Costa, Rhode Island College; Tristen Denley, University of Mississippi; Suzanne Feldberg, Nassau Community College; Margaret Finster, Erie Community College; Maryanne Frabotta, Community Campus of Beaver County; Lyn Geisler III, Randolph-Macon College; Patricia G. Granfield, George Mason University; Dale Grussing, Miami Dade College; Cindy Gubitose, Southern Connecticut State University; Virginia Harder, College at Oneonta; Joseph Lloyd Harris, Gulf Coast Community
College; Julia Hassett, Oakton Community College; Sonja Hensler, St. Petersburg College; James Henson, Edinboro University of Pennsylvania; Larry Hoehn, Austin Peay State University; Diane R. Hollister, Reading Area Community College; Kalynda Holton, Tallahassee Community College; Alec Ingraham, New Hampshire College; Linda Kuroski, Erie Community College—City Campus; Jamie Langille, University of Nevada, Las Vegas; Veronique Lanuqueitte, St. Petersburg College; Julia Ledet, Louisiana State University; Mitzi Logan, Pitt Community College; Dmitri Logvnenko, Phoenix College; Linda Lohman, Jefferson Community College; Richard J. Marchand, Slippery Rock University; Mike Marcozzi, University of Nevada, Las Vegas; Diana Martelly, Miami Dade College; Jim Matovina, Community College of Southern Nevada; Erik Matsuoka, Leeward Community College; Marcel Maupin, Oklahoma State University; Carrie McCammon, Ivy Tech Community College; Diana McCammon, Delgado Community College; Mex McKinley, Florida Keys Community College; Taranna Amani Miller, Indian River State College; Paul Mosbos, State University of New York—Cortland; Tammy Muhs, University of Central Florida; Cornelius Nelan, Quinnipiac University; Lawrence S. Orilia, Nassau Community College; Richard F. Patterson, University of North Florida; Frank Pecchioni, Jefferson Community College; Stan Perrine, Charleston Southern University; Anthony Pettofrezzo, University of Central Florida; Val Pinciu, Southern Connecticut State University; Evelyn Pupplo- Cody, Marshall University; Virginia S. Powell, University of Louisiana at Monroe; Kim Query, Lindenwood College; Anne Quinn, Edinboro University of Pennsylvania; Bill Quinn, Frederick Community College; Sharonda Ragland, ECPI College of Technology; Shawn Robinson, Valencia Community College; Gary Russell, Brevard Community College; Mary Lee Seitz, Erie Community College; Laurie A. Stahl, State University of New York—Fredonia; Abolhassan Taghavy, Richard J. Daley College & Chicago State University; Diane Tandy, New Hampshire Technical Institute; Ann Thrower, Kilgore College; Mike Tomme, Community College of Southern Nevada; Sherry Tornwall, University of Florida; Linda Tully, University of Pittsburgh at Johnstown; Christopher Scott Vaughen, Miami Dade College; Bill Vaughters, Valencia Community College; Karen Villareal, University of New Orleans; Don Warren, Edison Community College; Shirley Wilson, North Central College; James Wooland, Florida State University; Clifton E. Webb, Virginia Union University; Cindy Zarske, Fullerton College; Marilyn Zopp, McHenry County College
Additional acknowledgments are extended to
Brad Davis, for preparing the answer section and
annotated answers and serving as accuracy checker;
Bonnie Rosenblatt for writing the Learning Guide;
Acknowledgments
xii
Dan Miller and Kelly Barber, for preparing the solutions
manuals; the codeMantra formatting team for the book’s
brilliant paging; Brian Morris and Kevin Morris at
Scientific Illustrators, for superbly illustrating the book;
and Francesca Monaco, project manager, and Kathleen
Manley, production editor, whose collective talents kept
every aspect of this complex project moving through its
many stages.
I would like to thank my editors at Pearson, Dawn
Murrin and Marnie Greenhut, and editorial assistant,
Stacey Miller, who guided and coordinated the book from
manuscript through production. Finally, thanks to marketing
manager Kyle DiGiannantonio and marketing assistant
Brooke Imbornone for your innovative marketing efforts,
and to the entire Pearson sales force, for your confidence
and enthusiasm about the book.
Robert Blitzer
Acknowledgments xiii
xv
A
Activities, most-dreaded, 815–817
Actors, casting combinations, 698, 707, 765
Adjusted gross income, 504–505, 512–513,
575, 578
Advertisement, misleading, 159, 161–162
Affordable housing, voting on, 866, 894
Age
Americans’ definition of old age, 18–19
blood pressure and, 401–402
body-mass index and, 461
calculating, 262
car accidents and, 424, 488
of cars, on U.S. roads, 378
of Oscar winners, 784
of presidents, 783, 807, 841
stress level and, 436
Aging
body fat-to-muscle mass relationship
in, 28
near-light speed travel and, 299, 302
projected elderly population, 302
Airfares, 36–37
Alcohol
blood concentration of, 350, 353, 606
car accidents and, 472–473
Alligator, tail length of, 368
Ambassadors, seating arrangements
for, 930
Amortization schedule, 557–558, 577, 579
Angle(s)
of depression, from helicopter to
object, 675
of elevation
of kite string, 675
of Sun, 670–671, 674, 690
to top of Washington
Monument, 674
of wheelchair ramp, 675
of snow on windows, 624
on umbrellas, 623
Annuities, 530–532, 533, 542, 543,
553, 576, 579
Antimagic square, 41
Anxiety
in college students, 841
over dental work, 819
Apartments
option combinations, 699, 730
Applause levels, 434
Aquarium
volume of water in, 597–598, 600, 613
weight of water in, 604
Architecture
bidding for design, 761, 770
golden rectangles in, 298, 405
house length from scale, 38
Area
of islands, 601
of kitchen floor tiling, 655
to paint, 655
of rectangular room, 656
for shipping boxes, 690
Area codes, combinations of,
698, 699
Art, campaign posters as, 889
Awnings, 938
B
Baboon grooming behavior, 735–736
Ball(s). See also specific types of balls random selection of colored, 770
thrown height of, 483
Ballot measures, citizen-initiated, 869
Baseball, 591
batting orders, 703, 708
distance from home plate to second
base, 635
favorite players, 708
salaries in, 335
uniforms, loan to purchase, 518
weekly schedule, 906
Baseboard installation, 645, 688
Basketball, 39
dimensions of court, 644
free throw odds in favor, 743
volume of, 661
Berlin airlift (1948), 462, 467
Bicycle
hip angle of rider on, 624
manufacturing, 451
Bicycle-friendly communities, 409
Bike trail system, graphing, 938
Birthdays, probabilities and coincidence of
shared, 755
Births
per woman, contraceptives and, 836
worldwide, 378
Blood, red blood cells in the body, 340
Blood-alcohol concentration (BAC), 350,
353, 606
Blood drive, campus, 83, 99–100
Blood pressure, 401–402, 826
age and, 401–402
Blood transfusions, 94, 98
Body-mass index (BMI), 461
Book(s)
arrangement of, 701–702, 707, 765
book club selections, 713
collections of, 713
combinations of, 769
number read a year, 817
words read per minute, 38
Bookshelf manufacturing, 463, 464, 466
Box(es)
shipping, space needed by, 690
volume of, 664
Brain, growth of, 482
Breast cancer, mammography screening
for, 751–752
Budget deficit, federal, 339, 340
Buses
apportionment of, 873, 874–875,
876–877, 878, 881
fare options, 379
revenue from, 48
Business
branch location, 866
break-even point, 447–448, 450, 487
cocaine testing for employees, 723
cost of opening a restaurant, 47
customer service representatives, 714
defective products, 715
fractional ownership of franchise, 290
garage charges, 38
hamburger restaurant, 700
Internet marketing consultation, 704
investment in, 451
manufacturing costs, 353
officers, 707
profit, 39, 390, 488
maximization of, 466
promotions, 892, 895
revenue from bus operation, 48
self-employed’s workweek, 825
site selection, 762
C
Caloric needs, 346–347, 352
Campers, seating arrangements
for, 707, 714
Cancer, breast, 751–752
Canoe manufacturing, 451
Car(s)
accidents in
alcohol-related, 472–473
driver age and, 424, 488
outcome of, 754–755
average age of, on U.S. roads, 378
average annual costs of owning and
operating, 550, 553
average price of new, 378
depreciated value of, 39, 46, 378, 410
gasoline consumed, 47, 339
average gasoline prices, 153
Index of Applications
xvi Index of Applications
comparing fuel expenses, 550–551,
553, 577, 579
fuel efficiency, 47
supply and demand for unleaded
gasoline, 451
in a year, 38
loan on, 38, 546–547, 549–550,
552–553, 577
dealer incentives, 553
unpaid balance, 554
option combinations, 696–697, 698,
699, 769
rental cost, 39, 46, 380, 382, 389, 390
skidding distance and speed of, 301
stopping distance of, 417–418
tires, durability of, 841
Carbon dioxide in the atmosphere, 28
Cardiovascular disease, probability of, 741
Cards, probability of selecting, 718, 732,
734–735, 738–739, 741, 742, 743, 744,
748, 750–751, 753, 755, 766, 767, 770
Carpentry
baseboard costs, 645, 688
baseboard installation, 688
weekly salary, 17–18
Carpet installation, cost of, 647–648, 655,
656, 689
Casino gambling, opinions about, 773, 774
CD player, discount on, 497, 578
Cellphones
monthly charges for, 823
subscription to, 389
Cereals, potassium content of, 807
Certificate of deposit (CD),
517–518, 519
Checkout line, gender combinations
at, 708, 729
Child mortality, literacy and, 487, 842
Children, drug dosage for, 314
Chocolates, selection of, 747–748, 754,
766, 767
Cholesterol levels, 823, 842
Cigarette smoking. See Smoking City(-ies)
distance between, 591
ethnically diverse, 72
graph of, 906
hottest, 795
layout of, 40, 918, 941, 942, 944
with new college graduates, 798
New York City, 919
Real World, 866 snow removal, 125
visiting in random order, 766
Climate change, 28
Clock, movement around, in
degrees, 617
Clock addition, 310–311, 313
Club, officers of, 765
Coin toss, 720–721, 753, 769
College(s)
attendance at, 767
cost of, 44
election for president, 859
enrollment at university, 880
final course grade, 386–387, 389, 408,
776, 777, 796
professors
running for department chair, 857
running for division chair, 856
running for president of League of
Innovation, 856
room and board costs at, 482
College student(s)
anxiety in, 841
attitudes of, 372–374
binge drinking by, 107
careers most commonly named by
freshmen, 153
cigarette use by, 21–22
claiming no religious affiliation, 27
course registration, 108, 110
debt levels of, 29
emotional health of, 490
enrollment rates, 379
excuses for missing assignments, 289
and grade inflation, 47, 367
on greatest problems on campus, 12
heights of, 782
hours spent studying each
week, 844
IQ scores of, 783
majors of, 40
selection of, 768
musical styles preferred by, 108
participation in extracurricular
activities, 108
percent increase in lecture
registration, 575
random selection of freshmen vs. other
years, 749, 770
recruitment of male, 108
scholarships for minorities and
women, 107
selection of speakers by, 39, 862, 867
selection of topics by, 856
social interactions of, 782–783, 798
sources of news, 108
stress in, 782, 788, 791–792
symptoms of illness in procrastinators
vs. nonprocrastinators, 438, 451
time spent on homework, 782, 840
weight of male, 799
Color combinations, 98
Color printer, percent reduction from
original price, 502
Commercials, disclaimers in, 154
Committees
common members among, 906
formation of, 711, 713, 730, 766
Communication, monthly text message
plan, 46, 408, 410
Computer(s)
discounted sales price, 496–497
manufacturing, 491
payment time for, 48
quantum, 236
saving for, 38
Concerts, ordering of bands, 707, 708
Concrete, cost of, 665, 690
Condominium
property tax on, 502
purchase options, 765
Conference attendance, 714, 727–728, 729
Construction
affordable housing proposals, selecting,
866, 894
bidding on contract, 761, 769
of brick path, 646–647
carpet installation, 647–648, 655,
656, 689
costs of, 655, 656
of deck, 656
dirt removal, 665
of Great Pyramid, 665
kitchen floor tiling, 655
of new road, 636
pallets of grass, covering field with, 655
plastering, 655
residential solar installations, 483
of swimming pool, 658
tiling room, 655
trail in wilderness area, 645
trimming around window, 651
of wheelchair ramp, 632
Container, volume of, 600, 613
Contraceptives, births per woman and, 836
Cost(s)
of baseboard, 645, 688
of building new road, 636
of calculators, 27
of carpet, 647–648, 655, 656, 689
of ceramic tile, 656
of cigarette habit, 516–517
of college, 44
of college room and board, 482
comparison of, 38
of concrete, 665, 690
of construction, 656
of deck, 656
of fencing, 639, 645
of fertilizer, 655
to fill pool, 665
of gasoline, comparing, 550–551
of hauling dirt, 658, 665
of inflation, 407
of making a penny, 492
manufacturing, 353, 487
of oil pipeline, 656
for opening a restaurant, 47
Index of Applications xvii
pallets of grass, covering field
with, 655
of party, 40
per pound, 38
of pizza, 652, 656
of plastering, 655
of resurfacing path around swimming
pool, 656
of taxicab ride, 46
of tile installation, 655, 689
of tires, 38
of United States Census, 775
of vacation, 47
Counselors, school, 887–888
Countries, common borders between, 944
Creativity workshop, 290
Credit card(s)
average daily balance, 564–566, 570,
578, 580
balance owed on, 564–566, 578, 580
interest on, 564–566, 570, 578, 580
monthly payment on, 564–566, 570,
578, 580
Crowd, estimating number of people in, 17
D
Darts, 40, 723
Death and dying
infant, 842
involving firearms, 768, 832
leading causes of, 183
probability of dying at a given age, 724
worldwide, 378
Death-row inmates, final statements
of, 410
Debt
average U.S. household, 564
of college students, 29
national, 322–323, 325, 326
Decks, construction of, 656
Deficit, federal budget, 274–275, 339, 340
Delivery routes, 919
Delivery team, combinations of, 714
Demographics. See also Population Americans over 20 years old, 18–19
college graduates among people 25 and
older, 45
family composition, 722
life expectancy after 20, 743
literacy and child mortality, 487
living alone, 722
marital status, 720, 755
number of Americans who moved in
recent year, 723
Dentist, choosing, 42
Depression
exercise and, 436
humor and, 354, 360–361
treatments for, 109
Desk manufacturing, 463, 487
Dictionary, discounted price
for, 501, 575
Die/dice
expected value for roll of, 756,
762, 763
probability in rolling of, 716–717, 722,
738, 742, 743, 753, 755, 767
Diet. See Food Dimensions
of basketball court, 644
of football field, 644
of paper, 591
of rectangle, 644
Dinner party, guest arrivals, 729
Dinosaur walking speed, 339
Discount warehouse plans, 379
Disease(s)
sickle cell anemia, 722
Tay-Sachs, 766
tuberculosis, 768
Distance
across a lake, 674, 690
converting between mi/hr and
km/hr, 591
of helicopter from island, 674
from home plate to second base, 635
of ladder’s bottom from building, 688
rate and, 39
reach of ladder, 636
of ship from lighthouse, 675
of ship from shore, 674
of ship from Statue of Liberty, 674
sight, 301
between tracking stations, 636
traveled at given rate and time, 27
traveled by plane, 674
walking vs. jogging, 290
walking vs. riding bike, 39
Diversity index, 407
Doctors, apportionment of, 881, 895, 896
Documentaries, highest grossing, 72
Dogs, U.S. presidents with and
without, 83
Down payment on house, 534–535,
555–556, 561–562, 577, 579
saving for, 577
Dress, outfit combinations, 714
Drinks, combinations of orders, 699
Drivers. See also Car(s) ages of licensed, 827
intoxicated, on New Year’s Eve, 744
random selection of, 733–734
Driving, texting while, 490–491
Drug(s)
concentration of, 421
dosage, 598–599, 601, 605, 614
for children, 46, 314
nonprescription medications, 39
weight and, 610, 613
teenage use by country, 835, 837
E
Earnings
average yearly, by job, 378, 408
gender differences in, 437
from tutoring, 39, 467
weekly, 467, 818
Earthquake, on Richter scale, 317
Eating, hours and minutes per day spent
on, 791
Economics, 2009 stimulus package, 342
Education. See also College(s) bilingual math courses, 881
cost of attending a public
college, 23–25
department chairmanship, 857
final exam schedule, 856, 893
grants to states for, 342
home-schooling, 844
level of required, for jobs, 289
teacher-student ratio, 407
yearly earnings and, 370–372
Educational attainment
of 25-and-over population, 334,
723, 742
of college-graduate parents, 743
prejudice and, 831–832, 833, 834
Elections, 856, 859, 867, 896. See also Politics
mayoral, 857, 863–864, 896
probability of winning, 767, 770
Elevation, differences in, 274
Elevators, lifting capacity of, 389, 460
Employment. See also Job(s) in environmentally friendly
company, 841
as professor, 153
status of, 766
tree model of employee
relationships, 937
Enclosure(s)
fencing around circular garden, 689
of rectangular region, 656
Energy consumption, home energy pie, 19
English Channel tunnel, volume of dirt
removed to make, 665
Entertainment. See also Movies; Music; Television
play production, 451
Real World cities, 866 shared party costs, 40
theater revenue, 467
voting for play to perform, 856, 894
Environment, carbon dioxide in the
atmosphere, 28
Errands, route to run, 40, 929
Estate, division of, 290
Ethnicity
income by, 793
in police force, 767
in U.S. population, 329–330, 407
xviii Index of Applications
Examinations. See Test(s) Exercise
depression and, 436
maximum heart rate during, 352
Exercise machine, discounted
price, 501
Extraterrestrial intelligence, 261
Eye color, gender and, 770
F
Family, gender of children in, 745–746,
757, 767
FAX machine, discounted price for, 502
Fencing
around circular garden, 689
cost of, 639, 645
maximum area enclosed by, 656
Fertilizer, cost of, 655
Fiber-optic cable system, graphing,
938, 943
FICA taxes, 509, 513, 576, 578, 579
Finance. See Cost(s); Interest; Investment(s); Loan(s); Money; Mortgages
Firearms, deaths involving, 768, 832
Firefighter, rungs climbed by, 42
Fish pond, volume of, 598
Flagpole, cable supports for, 636
Flags, combinations of, 707
Flooding, probability of, 753, 768, 770
Floor plans, 683
connecting relationships in, 901–902,
907, 912–913, 919, 941, 944
Floor tiling, 655
Flu
HMO study of, 12
temperature curve during, 420–421
Flying time, time zones and, 46
Food
caloric needs, 346–347, 352
calories in hot dogs, 796–797
changing recipe size for preparing, 287,
290, 339
cholesterol-restricted diet, 461
estimating cost of meal, 17
supply and demand for packages of
cookies, 451
taste-testing, 860–861, 866
total spending on healthcare, 436
two-course meal, 765
Football
dimensions of field, 644
height of kicked ball, 351
height of thrown, 423
number of games required, 404
path of a punted, 478–479
in televised games, 732–733
401(k) plans, 540–541, 544
Frankfurters, amount for picnic, 46
Freshmen. See under College student(s)
Fund raiser, order of performance in, 729
Furnace capacity, 665
G
Game(s)
coin toss, 720–721, 753
darts, 40, 723
die rolling, 716–717, 722, 738, 742, 743,
753, 755, 767
expected value and, 759, 760, 770
numbers, 762
Scrabble tiles, 742–743, 754
Gardens
circular
enclosure of, 656
fencing around, 689
plants around, 656
flower bed, 645
Gender
best and worst places to be woman,
795–796
at checkout line, combinations of,
708, 729
of children in family, 745–746, 757, 767
earnings, gender differences in, 437
eye color and, 770
income by, 793
odds of randomly selecting male from
group, 770
police force and, 767
Genetics, cystic fibrosis and, 719
Government
budget surplus/deficit, 274–275
collection and spending of money by,
274–275. See also Tax(es) tax system, 158–159, 840
2009 economic stimulus package, 342
GPA, 799
Grade inflation in U.S. high schools, 47, 367
Greeting card venture, 451
Gross income, 504–505, 512–513, 575,
578, 579
Growth of boys, maximum yearly, 776, 778
Gun ownership, 409–410, 487
Gun violence, 820
Gym lockers, numbering of, 42
H
Hamachiphobia, 489
Happiness
during the day, 63
money and, 836
over time, 86
Health
aging and body fat, 28
emotional, of college freshmen, 490
exercise per week, 844
government-provided healthcare, 107
headaches per month, 844
panic attacks, 45
poverty and, 489
total spending on healthcare, 436
weight and, 457, 460, 461
weight ranges for given height,
367, 457
Health club plans, selecting, 378
Health indicators, worldwide, 97–98
Health insurance
premiums, 763
Health maintenance organization (HMO)
apportionment of doctors by, 881,
895, 896
flu study, 12
Heart rate, during exercise, 352
Height(s). See also Length of adults, 483, 808–812
of arch, 675
of building, 674, 675, 692
converting between meters and feet, 591
of eagle in flight, 490
of Eiffel Tower, 670
female, 824
femur length and, 368
healthy weight as function of, 460, 461
of kicked football, 351
of lamppost, 629, 688
median, 301
of plane, 675
of ramp, 636
of tower, 629, 632, 670, 674
of tree, 635, 674
weight and, 367, 457
High school students, most important
problems for, 27
Highway routes, 699
Hiking up slope, 690
Home(s). See also Mortgages affordable housing vote, 894
average size of, 781
down payment on, 534–535, 555–556,
561–562, 577, 579
saving for, 577
options available for new, 72
Homeless shelters, opinions about, 773
Home-schooling, 844
Homework, time spent on, 782, 840
Honeycombs, 638
Horse races, finishing combinations,
708, 740
Hospitalization, probability of, 755
Hot sauce, combinations of, 714
Humor, depression and, 354, 360–361
Hunger, literacy and, 836, 837
Hurricane, probability of, 746, 753
I
Ice cream, flavor combinations, 714
Illness, stress and, 830
Income
by gender and race, 793
Index of Applications xix
government’s responsibility for reducing
differences in, 102–103
of graduating college seniors, 13
gross, 504–505, 512–513, 575, 578, 579
taxable, 504–505, 512–513, 575, 578, 579
weekly earnings, 467, 818
Income tax. See Tax(es) Individual Retirement Accounts (IRAs),
533, 543, 576, 579
Infant deaths, 842
Infants, weight of, 812–813, 827
Insects, life cycles of, 261
Installment payment, on computer, 48
Insurance
automobile, 757–758
expected gain on policies sold, 762
premium on, 757–758
probabilities of claims, 761, 769
Intelligence, extraterrestrial, 261
Intelligence quotient. See IQ scores Interest, 576
on credit cards, 564–566
on investment, 579
on loans
compound, 527–528
simple, 514–515, 517, 518, 519
on mortgage, 559, 562, 577, 578, 580
on savings, 514–516, 520, 521–522,
527–528, 578
Inventiveness, beliefs about, 786–787
Investment(s)
accumulated value of, 527
in business venture, 451
choosing between, 522–523
gain and loss calculation, 502
of inheritance, 468
interest on, 579
lump-sum vs. periodic deposits, 543
present value of, 517–518
return on, 578, 805
for scholarship funds, 543
in stocks, 39, 699, 765, 805
percent increase/decrease, 575
price movements, 699, 765
return on, 805
share apportionment, 881
share purchase, 39
stock tables, 538–539, 542, 577, 579
volatility of, 807
IQ scores, 783, 814, 815, 819, 820, 844
Irrigation system, graphing, 938
J
Jacket, sale price of, 498–499
Japanese words, syllable frequency in, 842
Jet skis, 491
Job(s). See also Employment applicant qualifications, 153
applicant selections, 769
average yearly earnings by, 378, 408
comparing offers for, 335, 336
educational levels required for,
289–290
gender preferences for various, 85
opportunities for women vs. men, 114
shared night off from, 261
in U.S. solar-energy industry, 489
Job interview, turnoffs in, 820
Jogging
kilometers covered, 591, 612, 613
lapping other runner, 261
Jokes
combinations of, 701
ordering of, 714, 724–725
Juices, random selection of, 754
K
Königsberg, Germany, modeling, 899–900
L
Labor forces, Americans out of, 780
Lawns, fertilizer for, 655
Lawsuits
against contractor, 665
settlement vs., 762
Lectures on video, 339
Leisure activities, winter, 86
Length. See also Distance; Height(s) of alligator tail, 368
of blue whales, 587
of diplodocus, 588
of garden hose, 636
of trim around window, 651
Letters, combinations of, 706, 707, 715,
765, 769
License plate numbers and letters,
combinations of, 699
Life events, responding to negative,
360–361
Life expectancy, 20–21, 22–23, 268–269,
274, 502, 842
Literacy
child mortality and, 487, 842
hunger and, 836, 837
Literature, Shakespeare’s plays, 743
Loan(s). See also Interest car, 38, 546–547, 549–550, 552–553, 577
dealer incentives, 553
unpaid balance, 554
compounded interest on, 527–528
future value of, 516, 576
to pay off credit-card balance, 571
simple interest on, 514–515, 517, 518,
519, 576, 579
unpaid balance on, 565–566, 570
Logic problems, 42
Looks, distribution of, 150–151
Lottery(-ies), 713, 726–727
expected value in, 763
number selection for, 713, 715
probability of winning, 729, 766, 769
6/53, 715
Loudness, 489
Love
components of, 388–389
romantic, 125
M
Magic squares, 41
Mail routes, 902–903, 907, 918
Mail trucks, apportionment of, 892
Maintenance agreement, expected
profit per, 762
Mammography screening, 751–752
Map
legend of, 290
number of colors on, 40, 680
tracing route on, 40
Mapmaking, 671
Marital status, 720, 755
Marriage
between 20 to 24, 432
approval of equality in, sushi and,
835, 837
average age of first, 21
interfaith, 389
legal ages for, 175
romantic love as basis of, 125
Mass
atomic, 325
molecular, 325
Meals, combinations of courses, 695, 698,
699, 714
Medical volunteers, selection of, 713, 714
Memorabilia collectors, survey of, 104
Menendez trial, 188–189
Mental illness, U.S. adults with
serious, 489
Military, “don’t ask, don’t tell” policy, 47
Missing dollar problem, 42
Money
average price per movie ticket, 408
average price per rock concert
ticket, 341
cost of minting a penny, 492
dealer cost, 379
deferred payment plan, 376
digital camera price reduction, 375
division of, in will, 380
dollar’s purchasing power, 781
government collection and spending of,
274–275
happiness and, 836
lost wallet, 266
percent price decrease, 498–499, 502
price before reduction, 379, 380,
408, 410
sales commission, 408
sales tax, 379, 496–497, 501, 502, 575
stacking different denominations of, 261
xx Index of Applications
Money market account, 529
Mortgages, 555–556, 561–562
amortization schedule for,
557–558, 577
amount of, 577, 579
average rates, 559
comparing, 562, 577
cost of interest over term of, 562, 577
maximum affordable amount, 559–560,
578, 579
monthly payment on, 562, 577–578, 579
points at closing, 562, 577, 579
Movies
age distribution of moviegoers, 741
of Matthew McConaughey, 835, 837
with the most Oscar nominations, 98
order of showing, 769
Oscar winners, 784
rental options, 38–39
survey on, 103–105
theater times, 259, 261, 262
top rated, 72
top-rated documentaries, 707
viewing options, 72
Murder rates, 820–821
Music
choral group, 258, 261
college student preferences
for, 108
favorite CDs, 766
musical for new network, 857
note value and time signature, 290
order of performance of singers,
765, 766
platinum albums, 807
sounds created by plucked or bowed
strings, 290
stereo speakers selection, 861
survey on musical tastes, 100
top single recordings, 97
N
National park, area of, 593–594, 600, 612
Nature
honeycombs, 638
wilderness area, installation of trail
in, 645
New England states, common borders
among, 900–901, 919
Numbers
combinations of, 707, 708, 715, 766
palindromic, 723
Nursing staff, apportionment of, 881
O
Obesity, in mothers and daughters, 830
Oil pipeline, cost of, 656
Oscar awards, ages of winners, 784
Outfit combinations, 36
Overtime pay, 290
P
Painting, house, 655
Paper, dimensions of, 591
Paper manufacturing company, profit
margins, 488
Paragraphs, arrangement of sentences in,
706, 707
Parent-child relationships, tree model
of, 937
Parking space, combinations of
designations of, 699, 714
Passwords, four-letter, 713, 714, 715
Paths
brick, 646–647
resurfacing, 656
Payments
for computer, 48
credit card, 564–566
deferred plan, 376
in installment, 48
mortgage, 561–562, 577
Payoff periods, calculating, 33
Payroll, monthly, 44
Pens
choices of, 765
color of, 698
Pet ownership survey, 86
Photographs, arrangements of, 707
Pizza
combinations of orders, 699
cost of, 656
topping options, 72
Plane travel
runway line up, 766, 769
standbys selection, 713
Plastering, 655
Poker, possible 5-card hands,
711–712, 730
Poles, wires supporting, 688, 691
Police
apportionment among precincts, 881
ethnic and gender composition of, 767
patrol route, 920, 945
Police cars, dispatching options, 72
Police lineup, arrangements in, 706
Politics
campaign posters as art, 889
campaign promises, 499–500
city commissioners, 713, 765
committee formation, 712, 713,
714, 766
congressional seat allocation, 42
discussion group, 729, 754, 766
mayoral candidates, 854
mayoral election, 857, 863–864
ordinance
on nudity at public beaches, 867
on smoking, 866–867
president of the Student Film Institute,
848–850, 851–852, 853–854
probability of choosing one party over
another, 742
public support for jail
construction, 782
public support for school
construction, 782
Senate committee members, 713
Senate voting power, 870
state apportionment, 880–881, 882,
884–887, 891, 892, 895
student body president, 848
student president of club, 848
U.S. presidents
age of, 783, 801, 803, 807, 841
net worth of, 794, 798
Watergate scandal, 125
Pond, volume of, 598
Population. See also Demographics of bass in a lake, 368
of California, 335
of deer, 364
density of, 593, 600, 601, 612, 614
elderly, 302
of Florida, 341
of foreign-born Americans, 404–405
of fur seal pups, 368
of Greece, 379
growth, 332
projections, 48, 302
by state, 26
of Texas, 335
of trout in a lake, 407
of United States, 45, 302, 319–320,
324–325, 329–330, 332, 342
age 65 and over, 481–482
marital status of, 736–737, 755
percentage of high school graduates
and college graduates in,
433–434
of wildlife, 364, 410
of world, 45, 378, 470–472
projections through year 2150,
497–498
Poverty
attitudes about causes of, 101–102
health and, 489
rate of, 780
Pregnancies, lengths of, 824
Prejudice, educational attainment and,
831–832, 833, 834
Pressure, blood, 401–402, 826
age and, 401–402
Principal, selection of, 860
Prizes, ways of awarding, 765
Professors
ages of, 783
as mentors, 714
probability of choosing, vs.
instructor, 742
running for department chair, 857
Index of Applications xxi
running for division chair, 856
running for president of League of
Innovation, 856
Property
area of, 594–595, 600, 613, 614
tax on, 363–364
Public speaking, dread of, 815–817
Purchase, ways to receive change
for, 39, 40
Q
Quantum computers, 236
Questionnaires on student stress,
782, 788
R
Race(s)
finishing combinations,
35–36, 40, 707, 713
5 K, 608
income by, 793
lapping another racer, 261
Radio manufacturing, 450
Radio show, organization of, 707
Radio station call letters, combinations
of, 699
Raffles
award combinations, 713, 714
expected value of ticket
purchase, 760
odds against winning, 739, 743, 767
Rainfall, 591
Ramps
angle of elevation of, 675
height of, 636
Rapid transit service, 873, 874–875,
876–877, 878, 881
Real estate
appraisal of, 647
decision to list a house, 758
Recipes, changing size of, 287, 290, 339
Refrigerators, life of, 825
Relief supplies, distribution of, 261,
462–464, 465, 467
Religion
American adults believing in God,
Heaven, the devil, and Hell,
164–165
college students claiming no religious
affiliation, 27
Rental cost(s)
of boat, 48
of car, 39, 46
of movies, 38–39
Rescue from piranhas, 42
Retirement community, ages of people
living in, 841
Retirement planning, 528
401(k), 540–541, 544
IRAs, 533, 543, 544, 576
Return on investment, 805
on stocks, 805
Roads, inclined, 674
Rock concerts, average ticket price
for, 341
Roulette
expected value and, 760, 762
independent events on, 745
Rug cleaner, rental, 379
Rugs, length of fringe around
circular, 656
Running shoes, manufacturing, 448
S
Sailboat, area of sail on, 649
Salary(-ies)
after college, 353
annual increase in, 334, 335
baseball, 335
bonus to, 38
of carpenters, 17–18
and educational attainment,
370–372
of environmentally friendly
company, 841
mean vs. median, 792–793
of recent graduates, 783
reduction in, to work in environmentally
friendly company, 841
of salespeople, 844
of teachers, 44
wage gap by gender, 423
Sales director, traveling, 926, 928–929,
943, 945
Sales tax, 496–497, 501, 502, 575
Saving(s)
annuity value, 530–532, 533, 542, 543,
553, 576
for computer, 38
effective annual yield of, 524–526, 527,
528, 576, 579
interest on, 578
compound, 520, 521–522, 527, 529,
576, 579
simple, 576
present value of, 523
rate of, 334–335
for retirement, 528
IRAs, 533, 543, 544, 576, 579
for vacation, 543
Scheduling
of comedy acts, 704–705, 706,
714, 729
of night club acts, 706
by random selection, 729
of TV shows, 704–705, 707
Scholarship funds, 543
Scholarships for minorities and
women, 107
School courses. See also Education
combinations of, 695, 696
registration for, 108, 110
speed-reading, 799
School district
apportionment of counselors in,
887–888
laptops divided in, 891, 895
Scrabble tiles, 742–743, 754
Screens, measuring size of, 630–631
Seating arrangements, on
airplane, 708
Security guard, patrol route, 903, 907,
918, 942
Sex, legal age for, 175
Shaking hands, in groups, 40, 715
Shipping boxes, space needed by, 690
Shoes, combination with outfit, 695–696
Shopping
browsing time vs. amount spent
on, 489
for cans of soup, 665
categories of shoppers, 699
estimating total bill for, 17
unit price comparison, 31–32
Shower, water use during, 368
Sickle cell anemia, probability of
getting, 722
Sidewalks, clearing snow
from, 934–935
Sight distance, 301
Signs, triangular, 627
Simple interest, 576
on loan, 514–515, 517, 518, 519, 576, 579
on savings, 576
Skin, UV exposure of, 486
Sleep, average number of hours per
day, 791
by age, 63
Smoking
ailments associated
with, vs. nonsmoking, 109
alcohol and cigarette use by high school
seniors, 21–22
cost of habit, 516–517
ordinance on, 866–867
poll on, 107
Social Security, projected income and
outflow of, 410
Social Security numbers, combinations
of, 699
Society
American adults believing in God,
Heaven, the devil, and Hell,
164–165
class structure of the United States, 165
multilingual households, 82
social interactions of college students,
782–783, 798
women’s lives across continents and
cultures, 113
xxii Index of Applications
Solar power
number of jobs in U.S. solar-energy
industry, 489
residential installations, 483
Sound, intensity and loudness of, 489
Soups, ranking brands of, 857
Speed
converting between mi/hr and
km/hr, 589
of dinosaur walking, 339
skidding distance and, 301
Speed-reading course, 799
Spelling proficiency, 27
Spinner(s)
expected value for, 762, 769
probable outcomes in, 722, 736, 742, 753,
767, 770
Sports. See also specific sports intramural league, 257, 339
survey on winter activities people enjoy,
86, 115
Sports card collection, 261
States, common borders among, 901, 906,
919, 941
Stock(s), 39, 699, 765, 805
price movements of, 699, 765
return on investment in, 805
share apportionment, 881
share purchase, 39
volatility of, 807
Stock tables, 538–539, 542, 577, 579
Stonehenge, raising stone to build, 675
Stress
age and, 436
in college students, 782, 788, 791–792
illness and, 830
String instruments, sounds created by
plucked or bowed strings, 290
Students. See also College student(s) friendship pairs in homework
group, 906
studying time, 85
Subway system, London, 905
Sun
angle of elevation of, 670–671, 674, 690
distance from Earth to, 591
Surface area of cement block, 664
Swimming pool
construction of, 658
cost of filling, 665
volume of, 596, 600, 613, 614
T
Tattooed Americans, percentage of, 72
Tax(es)
deductions for home office, 655
FICA, 509, 513, 576, 578, 579
income, 502, 504–505, 513
computing, 507–508
federal, 507–508
net pay after, 511
withheld from gross pay, 510–511, 579
IRS fairness in, 158–159
marginal rates, 507–508, 512, 576, 578
percentage of work time spent paying
for, 502
percent reduction of, 499–500
property, 363–364
sales, 496–497, 501, 502, 575
state, 579
U.S. population and, 324–325
for working teen, 510–511, 513
Taxable income, 504–505, 512–513,
575, 578
Teachers, number required by school
board, 407
Teaching assistants, apportionment
of, 891
Telephone numbers, combinations
of, 697, 698, 699
Television
discount price, 575
football games on, 732–733
highest rated prime time shows on, 97
hours spent viewing, 29, 843
manufacturing, 467
M*A*S*H, viewership of final episode, 820
Nielsen Media Research
surveys, 820
NUMB3RS crime series, 288 percents misused on, 499
Roots, Part 8 viewership, 820 sale price, 499
screen measurement, 630–631
Temperature, 266
in enclosed vehicle, 474–475
estimating, 610
flu and, 420–421
perception of, 275
scale conversion, 351, 389, 438, 607, 609
Terminal illness, poll on, 108
Tessellations, 642, 644
Test(s)
ACT, 814
aptitude, 805
average score, 408
IQ, 783, 814, 819, 820, 844
multiple-choice, 697, 699, 765, 770
SAT, 759, 762, 814
scores on
comparing, 813–814
distribution of, 840, 841
frequency distribution for, 777
maximizing, 467
needed to achieve certain average,
408, 410
percentile, 844
stem-and-leaf plot for, 779
students classified by, 96–97
selection of questions and problems
in, 713
true/false, 40
Texting while driving, 490–491
Text message plan, monthly, 46, 408, 410
Tile installation, 691
cost of, 655, 689
Time
driving, 380
seconds in a year, 325
taken up counting, 27
to walk around road, 40
Toll(s)
discount pass for, 374, 379, 414–415
exact-change gates, 34–35
Transistors, defective, 729
Trash, amount of, 47
Travel club, voting on destination
city, 856
Treasury bills (T-bills), 519
Triangles, in signs, 627
Trip(s)
combinations of parts of, 699
selecting companions for, 748
Tuberculosis, 768
Tutoring, earnings for, 39, 467
U
Ultraviolet exposure, 486
University. See College(s) Unleaded gasoline, supply and demand
for, 451
V
Vacation, saving for, 543
Variety show, acts performed in,
765, 769
Vehicles. See Car(s) Vending machine, coin combinations for
45-cent purchase, 39
Volleyball tournament, elimination, 40
Volume
of basketball, 661
of box, 664
of car, 665
of cement block, 664
of cylinder, 664–665
of dirt from tunnel construction, 665
of Eiffel Tower, 665
of Great Pyramid, 665
of ice cream cone, 661
of pond, 598
of pyramid, 659, 690
Transamerica Tower, 659
of sphere, 664
Volunteers
for driving, 713
selection of, 714
Vowel, probability of
selecting, 750, 767
Index of Applications xxiii
W
Wages, overtime, 290. See also Salary(-ies) Washing machine, discounted price
for, 502
Water
gallons consumed while showering, 368
usage of, 665
utility charge for, 843
Water tank capacity, 665
Week, day of the, 42
Weight(s)
of adult men over 40, 842
drug dosage and, 610
estimating, 609
healthy ranges of, 367, 457, 460, 461
height and, 367, 457
of infants, 812–813, 827
of killer whale, 610
of male college students, 799
on moon, 368
Wheelchair
manufacturing, 447–448
ramps for, 632
Windows
stripping around stained glass, 656
trimming around, 651
Winter activities, survey of, 86, 115
Wood boards, sawing, 290
Words, longest, 790
Work, spending for average household
using 365 days worked, 502. See also Employment; Job(s)
Y
Yogurts, ranking brands of, 866
Z
Zoo, bear collections in, 712
Here’s where you’ll find these applications: Mathematical models involving college costs are developed in
Example 8 and Check Point 8 of Section 1.2. In Exercises 51
and 52 in Exercise Set 1.2, you will approach our climate
crisis mathematically by developing models for data related
to global warming.
Problem Solving and Critical Thinking 1
If these trends continue, what can we expect in the
2020s and beyond? We can answer this question by
using estimation techniques that allow us to represent
the data mathematically. With such representations,
called mathematical models, we can gain insights and
predict what might occur in the future on a variety of
issues, ranging from college costs to global warming.
HOW WOULD YOUR LIFESTYLE CHANGE IF A GALLON OF GAS COST $9.15?
OR IF THE PRICE OF A STAPLE SUCH AS MILK WAS $15? THAT’S HOW
much those products would cost if their prices had increased at the
same rate college tuition has increased since 1980.
TUITION AND FEES AT FOUR-YEAR COLLEGES
School Year
Ending 2000
School Year
Ending 2016
Public $3349 $9410
Private $15,518 $33,480
Source: The College Board
1
2 C H A P T E R 1 Problem Solving and Critical Thinking
A magnification of the Mandelbrot set
Richard F. Voss
ONE OF THE NEWER FRONTIERS OF MATHEMATICS SUGGESTS
that there is an underlying order in things that appear
to be random, such as the hiss and crackle of
background noises as you tune a radio.
Irregularities in the heartbeat, some of
them severe enough to cause a heart
attack, or irregularities in our
sleeping patterns, such as
insomnia, are examples of
chaotic behavior. Chaos
in the mathematical sense
does not mean a complete
lack of form or arrangement.
In mathematics, chaos is
used to describe something that
appears to be random but is not
actually random. The patterns of
chaos appear in images like the one
shown on the left, called the Mandelbrot
set. Magnified portions of this image yield
repetitions of the original structure, as well as
new and unexpected patterns. The Mandelbrot
set transforms the hidden structure of chaotic
events into a source of wonder and inspiration.
Many people associate mathematics with tedious computation, meaningless
algebraic procedures, and intimidating sets of equations. The truth is that
mathematics is the most powerful means we have of exploring our world and
describing how it works. The word mathematics comes from the Greek word mathematikos, which means “inclined to learn.” To be mathematical literally means to be inquisitive, open-minded, and interested in a lifetime of pursuing knowledge!
Mathematics and Your Life
A major goal of this book is to show you how mathematics can be applied to your life
in interesting, enjoyable, and meaningful ways. The ability to think mathematically
and reason with quantitative issues will help you so that you can:
• order and arrange your world by using sets to sort and classify information
(Chapter 2, Set Theory);
• use logic to evaluate the arguments of others and become a more effective
advocate for your own beliefs (Chapter 3, Logic);
• understand the relationship between cutting-edge technology and ancient
systems of number representation (Chapter 4, Number Representation and
Calculation);
• put the numbers you encounter in the news, from contemplating the national
debt to grasping just how colossal $1 trillion actually is, into perspective
(Chapter 5, Number Theory and the Real Number System);
• use mathematical models to gain insights into a variety of issues, including the
positive benefits that humor and laughter can have on your life (Chapter 6,
Algebra: Equations and Inequalities);
• use basic ideas about savings, loans, and investments to achieve your financial
goals (Chapter 8, Personal Finance);
• use geometry to study the shape of your world, enhancing your appreciation
of nature’s patterns and beauty (Chapter 10, Geometry);
• develop an understanding of the fundamentals of statistics and how these
numbers are used to make decisions (Chapter 12, Statistics);
1.1 Inductive and Deductive Reasoning WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Understand and use inductive reasoning.
2 Understand and use deductive reasoning.
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 3
• understand the mathematical paradoxes of voting in a democracy, increasing
your ability to function as a more fully aware citizen (Chapter 13, Voting and
Apportionment);
• use graph theory to examine how mathematics is used to solve problems in
the business world (Chapter 14, Graph Theory).
Mathematics and Your Career
Generally speaking, the income of an occupation is related to the amount of
education required. This, in turn, is usually related to the skill level required in
language and mathematics. With our increasing reliance on technology, the more
mathematics you know, the more career choices you will have.
Mathematics and Your World
Mathematics is a science that helps us recognize, classify, and explore the hidden
patterns of our universe. Focusing on areas as different as planetary motion, animal
markings, shapes of viruses, aerodynamics of figure skaters, and the very origin
of the universe, mathematics is the most powerful tool available for revealing the
underlying structure of our world. Within the last 40 years, mathematicians have
even found order in chaotic events such as the uncontrolled storm of noise in the
nerve cells of the brain during an epileptic seizure.
Inductive Reasoning
Mathematics involves the study of patterns. In everyday life, we frequently rely on
patterns and routines to draw conclusions. Here is an example:
The last six times I went to the beach, the traffic was light on Wednesdays and
heavy on Sundays. My conclusion is that weekdays have lighter traffic than
weekends.
This type of reasoning process is referred to as inductive reasoning, or induction.
“It is better to take what may seem to be too much math rather than too little. Career plans change, and one of the biggest roadblocks in undertaking new educational or training goals is poor preparation in mathematics. Furthermore, not only do people qualify for more jobs with more math, they are also better able to perform their jobs.” —Occupational Outlook Quarterly
1 Understand and use inductive reasoning.
I N D U C T I V E R E A S O N I N G
Inductive reasoning is the process of arriving at a general conclusion based on observations of specific examples.
Although inductive reasoning is a powerful method of drawing conclusions,
we can never be absolutely certain that these conclusions are true. For this reason,
the conclusions are called conjectures, hypotheses, or educated guesses. A strong inductive argument does not guarantee the truth of the conclusion, but rather provides
strong support for the conclusion. If there is just one case for which the conjecture
does not hold, then the conjecture is false. Such a case is called a counterexample.
EXAMPLE 1 Finding a Counterexample
The ten symbols that we use to write numbers, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and
9, are called digits. In each example shown below, the sum of two two-digit numbers is a three-digit number.
47 +73 120
56 +46 102
Is the sum of two two-digit numbers always a three-digit number? Find a
counterexample to show that the statement
The sum of two two-digit numbers is a three-digit number
is false.
4 C H A P T E R 1 Problem Solving and Critical Thinking
Here are two examples of inductive reasoning:
• Strong Inductive Argument In a random sample of 380,000 freshmen at 722 four-
year colleges, 25% said they frequently
came to class without completing readings
or assignments (Source: National Survey of Student Engagement). We can conclude
that there is a 95% probability that between
24.84% and 25.15% of all college freshmen
frequently come to class unprepared.
SOLUTION
There are many counterexamples, but we need to find only one. Here is an
example that makes the statement false:
56
+ 43 99
This example is a counterexample that shows the statement
The sum of two two-digit numbers is a three-digit number
is false.
Why is it so important to work each of the book’s Check Points?
You learn best by doing. Do
not simply look at the worked
examples and conclude that
you know how to solve them.
To be sure you understand
the worked examples, try
each Check Point. Check
your answer in the answer
section before continuing your
reading. Expect to read this
book with pencil and paper
handy to work the Check
Points.
GREAT QUESTION!
• Weak Inductive Argument Neither my dad nor my boyfriend has ever cried in
front of me. Therefore, men have difficulty
expressing their feelings.
Inductive reasoning is extremely important to mathematicians. Discovery in
mathematics often begins with an examination of individual cases to reveal patterns
about numbers.
EXAMPLE 2 Using Inductive Reasoning
Identify a pattern in each list of numbers. Then use this pattern to find the
next number.
a. 3, 12, 21, 30, 39, ______ b. 3, 12, 48, 192, 768, ______
c. 3, 4, 6, 9, 13, 18, ______ d. 3, 6, 18, 36, 108, 216, ______
SOLUTION
a. Because 3, 12, 21, 30, 39, ______ is increasing relatively slowly, let’s use addition as the basis for our individual observations.
+ = + = + = + =
3, 12, 21, 30, 39, _____
CHECK POINT 1 Find a counterexample to show that the statement The product of two two-digit numbers is a three-digit number
is false.
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 5
Generalizing from these observations, we conclude that each number
after the first is obtained by adding 9 to the previous number. Using this
pattern, the next number is 39 + 9, or 48. b. Because 3, 12, 48, 192, 768, ______ is increasing relatively rapidly, let’s
use multiplication as the basis for our individual observations.
× = × = × =
3, 12, 48, 192, 768, _____
× =
Generalizing from these observations, we conclude that each number
after the first is obtained by multiplying the previous number by 4.
Using this pattern, the next number is 768 * 4, or 3072. c. Because 3, 4, 6, 9, 13, 18, ______ is increasing relatively slowly, let’s use
addition as the basis for our individual observations.
3, 4, 6, 9, 13, 18, _____
+ = + = + = + = + =
Generalizing from these observations, we conclude that each number
after the first is obtained by adding a counting number to the previous
number. The additions begin with 1 and continue through each
successive counting number. Using this pattern, the next number is
18 + 6, or 24. d. Because 3, 6, 18, 36, 108, 216, ______ is increasing relatively rapidly, let’s
use multiplication as the basis for our individual observations.
3, 6, 18, 36, 108, 216, _____
× = × = × = × = × =
Generalizing from these observations, we conclude that each number
after the first is obtained by multiplying the previous number by 2 or by 3.
The multiplications begin with 2 and then alternate, multiplying by 2,
then 3, then 2, then 3, and so on. Using this pattern, the next number is
216 * 3, or 648.
“For thousands of years, people have loved numbers and found patterns and structures among them. The allure of numbers is not limited to or driven by a desire to change the world in a practical way. When we observe how numbers are connected to one another, we are seeing the inner workings of a fundamental concept.” —Edward B. Burger and Michael Starbird, Coincidences, Chaos, and All That Math Jazz, W. W. Norton and Company, 2005
CHECK POINT 2 Identify a pattern in each list of numbers. Then use this pattern to find the next number.
a. 3, 9, 15, 21, 27, ______
b. 2, 10, 50, 250, ______
c. 3, 6, 18, 72, 144, 432, 1728, ______
d. 1, 9, 17, 3, 11, 19, 5, 13, 21, ______
In our next example, the patterns are a bit more complex than the additions
and multiplications we encountered in Example 2.
EXAMPLE 3 Using Inductive Reasoning
Identify a pattern in each list of numbers. Then use this pattern to find the
next number.
a. 1, 1, 2, 3, 5, 8, 13, 21, ______ b. 23, 54, 95, 146, 117, 98, ______
6 C H A P T E R 1 Problem Solving and Critical Thinking
8
1
1
2
3
5
As this tree branches, the number of
branches forms the Fibonacci sequence.
SOLUTION
a. We begin with 1, 1, 2, 3, 5, 8, 13, 21. Starting with the third number in the list, let’s form our observations by comparing each number with the two
numbers that immediately precede it.
+ = + = + = + = + = + =
1, 1, 2, 3, 5, 8, 13, 21, _____
The first two numbers are 1. Generalizing from these observations, we
conclude that each number thereafter is the sum of the two preceding
numbers. Using this pattern, the next number is 13 + 21, or 34. (The numbers 1, 1, 2, 3, 5, 8, 13, 21, and 34 are the first nine terms of the
Fibonacci sequence, discussed in Chapter 5, Section 5.7.)
b. Now, we consider 23, 54, 95, 146, 117, 98. Let’s use the digits that form each number as the basis for our individual observations. Focus on the
sum of the digits, as well as the final digit increased by 1.
23, 54, 95, 146, 117, 98, _____
+ = + = + = + + = + + =
+ =+ =+ =+ =+ =
Generalizing from these observations, we conclude that for each number
after the first, we obtain the first digit or the first two digits by adding
the digits of the previous number. We obtain the last digit by adding 1
to the final digit of the preceding number. Applying this pattern to find
the number that follows 98, the first two digits are 9 + 8, or 17. The last digit is 8 + 1, or 9. Thus, the next number in the list is 179.
Can a list of numbers have more than one pattern?
Yes. Consider the illusion in Figure 1.1. This ambiguous figure contains two patterns, where it is not clear which pattern should predominate. Do you see a wine goblet or two faces looking at each other? Like this ambiguous figure, some lists of numbers
can display more than one pattern, particularly if only a few numbers are given. Inductive reasoning can result in more than one
probable next number in a list.
Example: 1, 2, 4, __________
Pattern: Each number after the first is obtained by multiplying the previous number by 2. The missing number is 4 * 2, or 8. Pattern: Each number after the first is obtained by adding successive counting numbers, starting with 1, to the previous number. The second number is 1 + 1, or 2. The third number is 2 + 2, or 4. The missing number is 4 + 3, or 7.
Inductive reasoning can also result in different patterns that produce the same probable
next number in a list.
Example: 1, 4, 9, 16, 25, __________
Pattern: Start by adding 3 to the first number. Then add successive odd numbers, 5, 7, 9, and so on. The missing number is 25 + 11, or 36. Pattern: Each number is obtained by squaring its position in the list: The first number is 12 = 1 * 1 = 1, the second number is 22 = 2 * 2 = 4, the third number is 32 = 3 * 3 = 9, and so on. The missing sixth number is 62 = 6 * 6, or 36.
The numbers that we found in Examples 2 and 3 are probable numbers. Perhaps you found patterns other than the ones we
pointed out that might have resulted in different answers.
F I G U R E 1 . 1
GREAT QUESTION!
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 7
CHECK POINT 3 Identify a pattern in each list of numbers. Then use this pattern to find the next number.
a. 1, 3, 4, 7, 11, 18, 29, 47, ______
b. 2, 3, 5, 9, 17, 33, 65, 129, ______
Mathematics is more than recognizing number patterns. It is about the patterns
that arise in the world around us. For example, by describing patterns formed
by various kinds of knots, mathematicians are helping scientists investigate the
knotty shapes and patterns of viruses. One of the weapons used against viruses
is based on recognizing visual patterns in the possible ways that knots can
be tied.
Our next example deals with recognizing visual patterns.
This electron microscope photograph
shows the knotty shape of the Ebola virus.
EXAMPLE 4 Finding the Next Figure in a Visual Sequence
Describe two patterns in this sequence of figures. Use the patterns to draw the
next figure in the sequence.
, , ,,
SOLUTION
The more obvious pattern is that the figures alternate between circles and
squares. We conclude that the next figure will be a circle. We can identify
the second pattern in the four regions containing no dots, one dot, two dots,
and three dots. The dots are placed in order (no dots, one dot, two dots, three
dots) in a clockwise direction. However, the entire pattern of the dots rotates
counterclockwise as we follow the figures from left to right. This means that
the next figure should be a circle with a single dot in the right-hand region,
two dots in the bottom region, three dots in the left-hand region, and no dots
in the top region.
The missing figure in the visual sequence, a circle with
a single dot in the right-hand region, two dots in the bottom
region, three dots in the left-hand region, and no dots in the
top region, is drawn in Figure 1.2.
F I G U R E 1 . 2
CHECK POINT 4 Describe two patterns in this sequence of figures. Use the patterns to draw the next figure in the sequence.
, , , ,
8 C H A P T E R 1 Problem Solving and Critical Thinking
Are You Smart Enough to Work at Google? In Are You Smart Enough to Work at Google? (Little, Brown, and Company, 2012), author William Poundstone guides readers
through the surprising solutions to challenging job-interview
questions. The book covers the importance of creative thinking
in inductive reasoning, estimation, and problem solving. Best of
all, Poundstone explains the answers.
Whether you’re preparing for a job interview or simply want to
increase your critical thinking skills, we highly recommend tackling
the puzzles in Are You Smart Enough to Work at Google? Here is a sample of two of the book’s problems that involve inductive
reasoning. We’ve provided hints to help you recognize the pattern
in each sequence. The answers appear in the answer section.
1. Determine the next entry in the sequence. SSS, SCC, C, SC, ______
Hint: Think of the capital letters in the English alphabet. A is made up of three straight lines. B consists of one straight
line and two curved lines. C is made up of one curved line.
2. Determine the next line in this sequence of digits.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
? ? ? ? ? ?
Blitzer Bonus
Deductive Reasoning
We use inductive reasoning in everyday life. Many of the conjectures that come
from this kind of thinking seem highly likely, although we can never be absolutely
certain that they are true. Another method of reasoning, called deductive reasoning, or deduction, can be used to prove that some conjectures are true.
D E D U C T I V E R E A S O N I N G
Deductive reasoning is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved to be true by deductive
reasoning is called a theorem.
Deductive reasoning allows us to draw a specific conclusion from one or more
general statements. Two examples of deductive reasoning are shown below. Notice
that in both everyday situations, the general statement from which the conclusion is
drawn is implied rather than directly stated.
?
2 Understand and use deductive reasoning.
• All proper names are prohibited in Scrabble. TEXAS is a proper name. Therefore, TEXAS is prohibited in Scrabble.
• All people need to sleep at 7 A.M. You sign up for a class at 7 A.M. Therefore, you'll sleep through the lecture or not even make it to class.
Deductive ReasoningEveryday Situation
One player to another in a Scrabble game: “You have to remove those five letters. You can’t use TEXAS as a word.”
Advice to college freshmen on choosing classes: “Never sign up for a 7 A.M. class. Yes, you did it in high school, but Mom was always there to keep waking you up, and if by some miracle you do make it to an early class, you will sleep through the lecture when you get there.”
(Source: How to Survive Your Freshman Year, Hundreds of Heads Books, 2004)
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 9
Our next example illustrates the difference between inductive and deductive
reasoning. The first part of the example involves reasoning that moves from specific
examples to a general statement, illustrating inductive reasoning. The second part of
the example begins with the general case rather than specific examples and illustrates
deductive reasoning. To begin the general case, we use a letter to represent any one of
various numbers. A letter used to represent any number in a collection of numbers is
called a variable. Variables and other mathematical symbols allow us to work with the general case in a very concise manner.
A BRIEF REVIEW In case you have forgotten
some basic terms of
arithmetic, the following list
should be helpful.
Sum: the result of
addition
Difference: the result of
subtraction
Product: the result of
multiplication
Quotient: the result of
division
T A B L E 1 . 1 Applying a Procedure to Four Individual Cases
Select a number. 4 7 11 100
Multiply the number by 6. 4 * 6 = 24 7 * 6 = 42 11 * 6 = 66 100 * 6 = 600
Add 8 to the product. 24 + 8 = 32 42 + 8 = 50 66 + 8 = 74 600 + 8 = 608
Divide this sum by 2. 32
2 = 16
50
2 = 25
74
2 = 37
608
2 = 304
Subtract 4 from the quotient. 16 - 4 = 12 25 - 4 = 21 37 - 4 = 33 304 - 4 = 300
EXAMPLE 5 Using Inductive and Deductive Reasoning
Consider the following procedure:
Select a number. Multiply the number by 6. Add 8 to the product. Divide
this sum by 2. Subtract 4 from the quotient.
a. Repeat this procedure for at least four different numbers. Write a conjecture that relates the result of this process to the original number selected.
b. Use the variable n to represent the original number and use deductive reasoning to prove the conjecture in part (a).
SOLUTION
a. First, let us pick our starting numbers. We will use 4, 7, 11, and 100, but we could pick any four numbers. Next we will apply the procedure given
in this example to 4, 7, 11, and 100, four individual cases, in Table 1.1.
Because we are asked to write a conjecture that relates the result of this
process to the original number selected, let us focus on the result of each case.
Original number selected 4 7 11 100
Result of the process 12 21 33 300
Do you see a pattern? Our conjecture is that the result of the process is
three times the original number selected. We have used inductive reasoning.
b. Now we begin with the general case rather than specific examples. We use the variable n to represent any number.
3n + 4 - 4 = 3n
6n + 8
6n (This means 6 times n.)
n
6n + 8 2
6n 2
8 2
= + = 3n + 4
Using the variable n to represent any number, the result is 3n, or three times the number n. This proves that the result of the procedure is three times the original number selected for any number. We have used
deductive reasoning. Observe how algebraic notation allows us to work
with the general case quite efficiently through the use of a variable.
10 C H A P T E R 1 Problem Solving and Critical Thinking
CHECK POINT 5 Consider the following procedure: Select a number. Multiply the number by 4. Add 6 to the product. Divide this
sum by 2. Subtract 3 from the quotient.
a. Repeat this procedure for at least four different numbers. Write a conjecture that relates the result of this process to the original number
selected.
b. Use the variable n to represent the original number and use deductive reasoning to prove the conjecture in part (a).
Surprising Friends with Induction Ask a few friends to follow this procedure:
Write down a whole number from 2 to 10. Multiply the
number by 9. Add the digits. Subtract 3. Assign a letter to this
result using A = 1, B = 2, C = 3, and so on. Write down the name of a state that begins with this letter. Select the name of
an insect that begins with the last letter of the state. Name a
fruit or vegetable that begins with the last letter of the insect.
After following this procedure, surprise your friend
by asking, “Are you thinking of an ant in Florida eating a
tomato?” (Try using inductive reasoning to determine how
you came up with this “astounding” question. Are other less-
probable “astounding” questions possible using inductive
reasoning?)
Blitzer Bonus
Fill in each blank so that the resulting statement is true.
1. The statement 3 + 3 = 6 serves as a/an ______________ to the conjecture that the sum of two odd numbers is an odd number.
2. Arriving at a specific conclusion from one or more general statements is called ___________ reasoning.
3. Arriving at a general conclusion based on observations of specific examples is called ___________ reasoning.
4. True or False: A theorem cannot have counterexamples. _______
Concept and Vocabulary Check
Exercise Set 1.1
Any way that I can perk up my brain before working the book’s Exercise Sets?
Researchers say the mind can be strengthened, just like your muscles, with regular training
and rigorous practice. Think of the book’s Exercise Sets as brain calisthenics. If you’re feeling a
bit sluggish before any of your mental workouts, try this warmup:
In the list below, say the color the word is printed in, not the word itself. Once you can
do this in 15 seconds without an error, the warmup is over and it’s time to move on to
the assigned exercises.
Blue Yellow Red Green Yellow Green Blue Red Yellow Red
GREAT QUESTION!
What am I supposed to do with the exercises in the Concept and Vocabulary Check?
An important component of thinking mathematically involves knowing the special language and notation used in mathematics.
The exercises in the Concept and Vocabulary Check, mainly fill-in-the-blank and true/false items, test your understanding of the
definitions and concepts presented in each section. Work all of the exercises in the Concept and Vocabulary Check regardless of which exercises your professor assigns in the Exercise Set that follows.
GREAT QUESTION!
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 11
Practice Exercises
In Exercises 1–8, find a counterexample to show that each of the statements is false.
1. No U.S. president has been younger than 65 at the time of his inauguration.
2. No singers appear in movies.
3. If a number is multiplied by itself, the result is even.
4. The sum of two three-digit numbers is a four-digit number.
5. Adding the same number to both the numerator and the denominator (top and bottom) of a fraction does not change
the fraction’s value.
6. If the difference between two numbers is odd, then the two numbers are both odd.
7. If a number is added to itself, the sum is greater than the original number.
8. If 1 is divided by a number, the quotient is less than that number.
In Exercises 9–38, identify a pattern in each list of numbers. Then use this pattern to find the next number. (More than one pattern might exist, so it is possible that there is more than one correct answer.)
9. 8, 12, 16, 20, 24, ______ 10. 19, 24, 29, 34, 39, ______
11. 37, 32, 27, 22, 17, ______ 12. 33, 29, 25, 21, 17, ______
13. 3, 9, 27, 81, 243, _______ 14. 2, 8, 32, 128, 512, ______
15. 1, 2, 4, 8, 16, ______ 16. 1, 5, 25, 125, _______
17. 1, 4, 1, 8, 1, 16, 1, ______ 18. 1, 4, 1, 7, 1, 10, 1, ______
19. 4, 2, 0, - 2, - 4, _______ 20. 6, 3, 0, - 3, - 6, _____ 21. 12 ,
1 6 ,
1 10 ,
1 14 ,
1 18 , _____ 22. 1,
1 2 ,
1 3 ,
1 4 ,
1 5 , _____
23. 1, 13 , 1 9 ,
1 27 , _____ 24. 1,
1 2 ,
1 4 ,
1 8 , _____
25. 3, 7, 12, 18, 25, 33, ______ 26. 2, 5, 9, 14, 20, 27, ______
27. 3, 6, 11, 18, 27, 38, ______ 28. 2, 5, 10, 17, 26, 37, ______
29. 3, 7, 10, 17, 27, 44, ______ 30. 2, 5, 7, 12, 19, 31, ______
31. 2, 7, 12, 5, 10, 15, 8, 13, ______
32. 3, 9, 15, 5, 11, 17, 7, 13, ______
33. 3, 6, 5, 10, 9, 18, 17, 34, ______
34. 2, 6, 5, 15, 14, 42, 41, 123, _______
35. 64, - 16, 4, - 1, _____ 36. 125, - 25, 5, - 1, _____ 37. (6, 2), (0, - 4), 17 12 , 3
1 22, (2, - 2), (3, ______ )
38. 123 , 4 92, 1
1 5 ,
1 252, (7, 49), 1 -
5 6 ,
25 362, 1 -
4 7 , ______2
In Exercises 39–42, identify a pattern in each sequence of figures. Then use the pattern to find the next figure in the sequence.
39.
, , , ,,
40.
, ,, , ,
41.
,
a a
,
b b b
,
c
c
c c
42.
, , , ,
Exercises 43–46 describe procedures that are to be applied to numbers. In each exercise,
a. Repeat the procedure for four numbers of your choice. Write a conjecture that relates the result of the process to the original number selected.
b. Use the variable n to represent the original number and use deductive reasoning to prove the conjecture in part (a).
43. Select a number. Multiply the number by 4. Add 8 to the product. Divide this sum by 2. Subtract 4 from the
quotient.
44. Select a number. Multiply the number by 3. Add 6 to the product. Divide this sum by 3. Subtract the original selected
number from the quotient.
45. Select a number. Add 5. Double the result. Subtract 4. Divide by 2. Subtract the original selected number.
46. Select a number. Add 3. Double the result. Add 4. Divide by 2. Subtract the original selected number.
In Exercises 47–52, use inductive reasoning to predict the next line in each sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct.
47. 1 + 2 = 2 * 3
2
1 + 2 + 3 = 3 * 4
2
1 + 2 + 3 + 4 = 4 * 5
2
1 + 2 + 3 + 4 + 5 = 5 * 6
2
48. 3 + 6 = 6 * 3
2
3 + 6 + 9 = 9 * 4
2
3 + 6 + 9 + 12 = 12 * 5
2
3 + 6 + 9 + 12 + 15 = 15 * 6
2
49. 1 + 3 = 2 * 2 1 + 3 + 5 = 3 * 3
1 + 3 + 5 + 7 = 4 * 4 1 + 3 + 5 + 7 + 9 = 5 * 5
50. 1 * 9 + (1 + 9) = 19 2 * 9 + (2 + 9) = 29 3 * 9 + (3 + 9) = 39 4 * 9 + (4 + 9) = 49
51. 9 * 9 + 7 = 88 98 * 9 + 6 = 888
987 * 9 + 5 = 8888 9876 * 9 + 4 = 88,888
52. 1 * 9 - 1 = 8 21 * 9 - 1 = 188
321 * 9 - 1 = 2888 4321 * 9 - 1 = 38,888
12 C H A P T E R 1 Problem Solving and Critical Thinking
Practice Plus
In Exercises 53–54, use inductive reasoning to predict the next line in each sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct.
53. 33 * 3367 = 111,111 66 * 3367 = 222,222 99 * 3367 = 333,333
132 * 3367 = 444,444 54. 1 * 8 + 1 = 9
12 * 8 + 2 = 98 123 * 8 + 3 = 987
1234 * 8 + 4 = 9876 12,345 * 8 + 5 = 98,765
55. Study the pattern in these examples: a2 # a4 = a10 a3 # a2 = a7 a5 # a3 = a11.
Select the equation that describes the pattern.
a. ax # ay = a2x + y b. ax # ay = ax + 2y
c. ax # ay = ax + y + 4 d. ax # ay = axy + 2
56. Study the pattern in these examples: a5 * a3 * a2 = a5 a3 * a7 * a2 = a6 a2 * a4 * a8 = a7.
Select the equation that describes the pattern.
a. ax * ay * az = ax + y + z b. ax * ay * az = a xyz 2
c. ax * ay * az = a x + y + z
2 d. ax * ay * az = a xy 2
+ z
Application Exercises
In Exercises 57–60, identify the reasoning process, induction or deduction, in each example. Explain your answer.
57. It can be shown that
1 + 2 + 3 + g + n = n(n + 1)
2 .
I can use this formula to conclude that the sum of the first
one hundred counting numbers, 1 + 2 + 3 + g + 100, is 100(100 + 1)
2 =
100(101)
2 = 50(101), or 5050.
58. An HMO does a follow-up study on 200 randomly selected patients given a flu shot. None of these people became
seriously ill with the flu. The study concludes that all HMO
patients be urged to get a flu shot in order to prevent a
serious case of the flu.
59. The data in the graph are from a random sample of 1200 full- time four-year undergraduate college students on 100 U.S.
campuses.
40%
50%
30%
10%
20%
P e rc
e n
ta ge
o f
S tu
d e n
ts Id
e n
ti fy
in g
th e P
ro b
le m
The Greatest Problems on Campus
Alcohol Abuse
44%
Cost of
Education
40%
Student Loan Debt
23%
Lack of Financial
Aid
21%
Drug Abuse
19%
Drunk Driving
18%
Source: Student Monitor LLC
Using the graph at the bottom of the previous column, we can
conclude that there is a high probability that approximately
44% of all full-time four-year college students in the United
States believe that alcohol abuse is the greatest problem on
campus.
60. The course policy states that work turned in late will be marked down a grade. I turned in my report a day late, so it was marked
down from B to C.
61. The ancient Greeks studied figurate numbers, so named because of their representations as geometric arrangements
of points.
Triangular Numbers
1 3 6 10 15 21
Square Numbers
1 4 9 16 25
Pentagonal Numbers
1 5 12 22
a. Use inductive reasoning to write the five triangular numbers that follow 21.
b. Use inductive reasoning to write the five square numbers that follow 25.
c. Use inductive reasoning to write the five pentagonal numbers that follow 22.
d. Use inductive reasoning to complete this statement: If a triangular number is multiplied by 8 and then 1 is added
to the product, a _______ number is obtained.
62. The triangular arrangement of numbers shown below is known as Pascal’s triangle, credited to French mathematician Blaise Pascal (1623–1662). Use inductive reasoning to find
the six numbers designated by question marks.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
? ? ? ? ? ?
Explaining the Concepts
An effective way to understand something is to explain it to someone else. You can do this by using the Explaining the Concepts exercises that ask you to respond with verbal or written explanations. Speaking about a new concept uses a different part of your brain than thinking about the concept. Explaining new ideas verbally will quickly reveal any gaps in your understanding. It will also help you to remember new concepts for longer periods of time.
63. The word induce comes from a Latin term meaning to lead. Explain what leading has to do with inductive reasoning.
64. Describe what is meant by deductive reasoning. Give an example.
S E C T I O N 1 . 1 Inductive and Deductive Reasoning 13
65. Give an example of a decision that you made recently in which the method of reasoning you used to reach
the decision was induction. Describe your reasoning
process.
Critical Thinking Exercises
Make Sense? In Exercises 66–69, determine whether each statement makes sense or does not make sense, and explain your reasoning.
66. I use deductive reasoning to draw conclusions that are not certain, but likely.
67. Additional information may strengthen or weaken the probability of my inductive arguments.
68. I used the data shown in the bar graph, which summarizes a random sample of 752 college seniors, to conclude with
certainty that 51% of all graduating college females expect
to earn $30,000 or less after graduation.
60%
50%
40%
30%
20%
P er
ce n
ta ge
o f
G ra
d u
at in
g C
o ll
eg e
S en
io rs
First-Year Income Expectations of Graduating College Seniors
$50,000 or more
$30,000 or less
12%
29%
51%
35%
10%
Men
Women
Source: Duquesne University Seniors’ Economic Expectation Research Survey
69. I used the data shown in the bar graph for Exercise 68, which summarizes a random sample of 752 college seniors,
to conclude inductively that a greater percentage of male
graduates expect higher first-year income than female
graduates.
70. If (6 - 2)2 = 36 - 24 + 4 and (8 - 5)2 = 64 - 80 + 25, use inductive reasoning to write a compatible expression for
(11 - 7)2. 71. The rectangle shows an array of nine numbers represented
by combinations of the variables a, b, and c.
a + b a - b - c a + c
a - b + c a a + b - c
a - c a + b + c a - b
a. Determine the nine numbers in the array for a = 10, b = 6, and c = 1. What do you observe about the sum of the numbers in all rows, all columns, and the two
diagonals?
b. Repeat part (a) for a = 12, b = 5, and c = 2. c. Repeat part (a) for values of a, b, and c of your choice.
d. Use the results of parts (a) through (c) to make an inductive conjecture about the rectangular array of nine
numbers represented by a, b, and c.
e. Use deductive reasoning to prove your conjecture in part (d).
72. Write a list of numbers that has two patterns so that the next number in the list can be 15 or 20.
73. a. Repeat the following procedure with at least five people. Write a conjecture that relates the result of the
procedure to each person’s birthday.
Take the number of the month of your birthday
(January = 1, February = 2, c , December = 12), multiply by 5, add 6, multiply this sum by 4, add 9,
multiply this new sum by 5, and add the number of the
day on which you were born. Finally, subtract 165.
b. Let M represent the month number and let D represent the day number of any person’s birthday. Use deductive
reasoning to prove your conjecture in part (a).
Technology Exercises
74. a. Use a calculator to find 6 * 6, 66 * 66, 666 * 666, and 6666 * 6666.
b. Describe a pattern in the numbers being multiplied and the resulting products.
c. Use the pattern to write the next two multiplications and their products. Then use your calculator to verify
these results.
d. Is this process an example of inductive or deductive reasoning? Explain your answer.
75. a. Use a calculator to find 3367 * 3, 3367 * 6, 3367 * 9, and 3367 * 12.
b. Describe a pattern in the numbers being multiplied and the resulting products.
c. Use the pattern to write the next two multiplications and their products. Then use your calculator to verify
these results.
d. Is this process an example of inductive or deductive reasoning? Explain your answer.
Group Exercise
76. Stereotyping refers to classifying people, places, or things according to common traits. Prejudices and stereotypes
can function as assumptions in our thinking, appearing
in inductive and deductive reasoning. For example, it is
not difficult to find inductive reasoning that results in
generalizations such as these, as well as deductive reasoning
in which these stereotypes serve as assumptions:
School has nothing to do with life.
Intellectuals are nerds.
People on welfare are lazy.
Each group member should find one example of inductive
reasoning and one example of deductive reasoning in which
stereotyping occurs. Upon returning to the group, present
each example and then describe how the stereotyping
results in faulty conjectures or prejudging situations and
people.
14 C H A P T E R 1 Problem Solving and Critical Thinking
1.2 Estimation, Graphs, and Mathematical Models WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Use estimation techniques to arrive at an approximate answer
to a problem.
2 Apply estimation techniques to information given by graphs.
3 Develop mathematical models that estimate relationships
between variables.
1 Use estimation techniques to arrive at an approximate answer to a problem.
IF PRESENT TRENDS CONTINUE, IS IT POSSIBLE THAT OUR DESCENDANTS COULD LIVE
to be 200 years of age? To answer this question, we need to examine data for
life expectancy and develop estimation techniques for representing the data
mathematically. In this section, you will learn estimation methods that will enable
you to obtain mathematical representations of data displayed by graphs, using
these representations to predict what might occur in the future.
Estimation
Estimation is the process of arriving at an approximate answer to a question. For example, companies estimate the amount of their products consumers are likely to
use, and economists estimate financial trends. If you are about to cross a street, you
may estimate the speed of oncoming cars so that you know whether or not to wait
before crossing. Rounding numbers is also an estimation method. You might round
a number without even being aware that you are doing so. You may say that you
are 20 years old, rather than 20 years 5 months, or that you will be home in about a
half-hour, rather than 25 minutes.
You will find estimation to be equally valuable in your work for this class.
Making mistakes with a calculator or a computer is easy. Estimation can tell us
whether the answer displayed for a computation makes sense.
In this section, we demonstrate several estimation methods. In the second part
of the section, we apply these techniques to information given by graphs.
Rounding Whole Numbers
The numbers that we use for counting, 1, 2, 3, 4, 5, 6, 7, and so on, are called
natural numbers. When we combine 0 with the natural numbers, we obtain the whole numbers.
W H O L E N U M B E R S
The whole numbers are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … .
The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits, from the Latin word for fingers. Digits are used to write whole numbers.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 15
The position of each digit in a whole number tells us the value of that digit. Here
is an example using world population at 7:35 a.m. Eastern Time on January 9, 2017.
7 , 4 7 6 , 2 4 2 , 0 5 6
When do I need to use hyphens to write the names of numbers?
Hyphenate the names for
the numbers 21 (twenty-one)
through 99 (ninety-nine),
except 30, 40, 50, 60, 70, 80,
and 90.
GREAT QUESTION!
R O U N D I N G W H O L E N U M B E R S
1. Look at the digit to the right of the digit where rounding is to occur.
2. a. If the digit to the right is 5 or greater, add 1 to the digit to be rounded. Replace all digits to the right with zeros.
b. If the digit to the right is less than 5, do not change the digit to be rounded. Replace all digits to the right with zeros.
The symbol ≈ means is approximately equal to. We will use this symbol when rounding numbers.
EXAMPLE 1 Rounding a Whole Number
Round world population (7,476,242,056) as follows:
a. to the nearest hundred-million b. to the nearest million c. to the nearest hundred-thousand.
SOLUTION
a. 7,476,242,056 L 7,500,000,000
World population to the nearest hundred-million is seven billion,
five hundred-million.
b. 7,476,242,056 L 7,476,000,000
World population to the nearest million is seven billion, four hundred
seventy-six million.
c. 7,476,242,056 L 7,476,200,000
World population to the nearest hundred-thousand is seven billion, four
hundred seventy-six million, two hundred thousand.
16 C H A P T E R 1 Problem Solving and Critical Thinking
Rounding can also be applied to decimal notation, used to denote a part
of a whole. Once again, the place that a digit occupies tells us its value. Here’s
an example using the first seven digits of the number p (pi). (We’ll have more
to say about p, whose digits extend endlessly with no repeating pattern, in
Chapter 5.)
p L 3 . 1 4 1 5 9 2
We round the decimal part of a decimal number in nearly the same way that we
round whole numbers. The only difference is that we drop the digits to the right of
the rounding place rather than replacing these digits with zeros.
CHECK POINT 1 Round world population (7,476,242,056) as follows: a. to the nearest billion
b. to the nearest ten-million.
CHECK POINT 2 Round 3.141592, the first seven digits of p, as follows: a. to the nearest tenth
b. to the nearest ten-thousandth.
EXAMPLE 2 Rounding the Decimal Part of a Number
Round 3.141592, the first seven digits of p, as follows:
a. to the nearest hundredth b. to the nearest thousandth.
SOLUTION
a. 3.141592 L 3.14
The number p to the nearest hundredth is three and fourteen hundredths.
b. 3.141592 L 3.142
The number p to the nearest thousandth is three and one hundred
forty-two thousandths.
Could you please explain how the decimal numbers in Example 2 are read?
Of course! The whole-number
part to the left of the decimal
point is read like any whole
number, which is three in both parts of Example 2. The
decimal point is read as and. The decimal part to the right of
the decimal point is read like a
whole number followed by the
place value of the rightmost
digit. In 3.14, the 4 is in the
hundredths place, so there are
fourteen hundredths. In 3.142, the 2 is in the thousandths
place, so there are one hundred forty-two thousandths.
GREAT QUESTION!
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 17
EXAMPLE 3 Estimation by Rounding
You purchased bread for $2.59, detergent for $5.17, a sandwich for $3.65, an
apple for $0.47, and coffee for $8.79. The total bill was given as $24.67. Is this
amount reasonable?
SOLUTION
If you are in the habit of carrying a calculator to the store, you can answer the
question by finding the exact cost of the purchase. However, estimation can be
used to determine if the bill is reasonable even if you do not have a calculator.
We will round the cost of each item to the nearest dollar.
Bread $2.59 L $3.00 Detergent $5.17 L $5.00 Sandwich $3.65 L $4.00 Apple $0.47 L $0.00 Coffee $8.79 L $9.00
$21.00
The total bill that you were given, $24.67, seems a bit high compared to the
$21.00 estimate. You should check the bill before paying it. Adding the prices
of all five items gives the true total bill of $20.67.
CHECK POINT 3 You and a friend ate lunch at Ye Olde Cafe. The check for the meal showed soup for $3.40, tomato juice for $2.25, a roast beef sandwich for
$5.60, a chicken salad sandwich for $5.40, two coffees totaling $3.40, apple pie
for $2.85, and chocolate cake for $3.95.
a. Round the cost of each item to the nearest dollar and obtain an estimate for the food bill.
b. The total bill before tax was given as $29.85. Is this amount reasonable?
EXAMPLE 4 Estimation by Rounding
A carpenter who works full time earns $28 per hour.
a. Estimate the carpenter’s weekly salary. b. Estimate the carpenter’s annual salary.
SOLUTION
a. In order to simplify the calculation, we can round the hourly rate of $28 to $30. Be sure to write out the units for each number in the calculation.
The work week is 40 hours per week, and the rounded salary is $30 per
hour. We express this as
40 hours
week and
+30 hour
.
Police often need to estimate
the size of a crowd at a political
demonstration. One way to do
this is to select a reasonably
sized rectangle within the crowd
and estimate (or count) the
number of people within the
rectangle. Police then estimate
the number of such rectangles
it would take to completely
fill the area occupied by the
crowd. The police estimate is
obtained by multiplying the
number of such rectangles by
the number of demonstrators in
the representative rectangle. The
org anizers of the demonstration
might give a larger estimate
than the police to emphasize the
strength of their support.
Blitzer Bonus Estimating Support for a Cause
18 C H A P T E R 1 Problem Solving and Critical Thinking
The word per is represented by the division bar. We multiply these two numbers to estimate the carpenter’s weekly salary. We cancel out units
that are identical if they are above and below the division bar.
40 hours
week * +30
hour = +1200 week
Thus, the carpenter earns approximately $1200 per week, written
≈+1200. b. For the estimate of annual salary, we may round 52 weeks to 50 weeks.
The annual salary is approximately the product of $1200 per week and
50 weeks per year:
+1200 week
* 50 weeks
year = +60,000
year .
Thus, the carpenter earns approximately $60,000 per year, or $60,000
annually, written ≈+60,000.
CHECK POINT 4 A landscape architect who works full time earns $52 per hour. a. Estimate the landscape architect’s weekly salary.
b. Estimate the landscape architect’s annual salary.
Is it OK to cancel identical units if one unit is singular and the other is plural?
Yes. It does not matter whether
a unit is singular, such as week, or plural, such as weeks. Week and weeks are identical units and can be canceled out, as
shown on the right.
GREAT QUESTION!
Estimation with Graphs
Magazines, newspapers, and
websites often display information
using circle, bar, and line graphs.
The following examples illustrate
how rounding and other estimation
techniques can be applied to data
displayed in each of these types of
graphs.
Circle graphs, also called pie charts, show how a whole quantity is divided into parts. Circle graphs are
divided into pieces, called sectors. Figure 1.3 shows a circle graph that indicates how Americans disagree
as to when “old age” begins.
2 Apply estimation techniques to information given by graphs. Americans’ Definition of Old Age
Decline in physical ability
Becoming a grandparent
Retirement
Don’t know
Reaching a specific age
Decline in mental functioning
32%
41%
3%
1% 9%
14%
F I G U R E 1 . 3
Source: American Demographics
A BRIEF REVIEW Percents • Percents are the result of expressing numbers as part of 100. The word percent
means per hundred. For example, the circle graph in Figure 1.3 shows that 41% of Americans define old age by a decline in physical ability. Thus, 41 out of every
100 Americans define old age in this manner: 41, = 41100. • To convert a number from percent form to decimal form, move the decimal point two
places to the left and drop the percent sign. Example:
41% = 41.% = 0.41%
Thus, 41, = 0.41. • Many applications involving percent are based on the following formula:
A P
A = P ∙ B.
B
Note that the word of implies multiplication.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 19
In our next example, we will use the information in the circle graph on page 18
to estimate a quantity. Although different rounding results in different estimates,
the whole idea behind the rounding process is to make calculations simple.
EXAMPLE 5 Applying Estimation Techniques to a Circle Graph
According to the U.S. Census Bureau, in 2016, there were 219,345,624
Americans 25 years and older. Assuming the circle graph in Figure 1.3 is representative of this age group,
a. Use the appropriate information displayed by the graph to determine a calculation that shows the number of Americans 25 years and older who
define old age by a decline in physical ability.
b. Use rounding to find a reasonable estimate for this calculation.
SOLUTION
a. The circle graph in Figure 1.3 indicates that 41% of Americans define old age by a decline in physical ability. Among the 219,345,624 Americans
25 years and older, the number who define old age in this manner is
determined by finding 41% of 219,345,624.
= 0.41 * 219,345,624
b. We can use rounding to obtain a reasonable estimate of 0.41 * 219,345,624.
0.41 * 219,345,624 L 0.4 * 220,000,000 = 88,000,000
×
Our answer indicates that approximately 88,000,000 (88 million)
Americans 25 years and older define old age by a decline in physical
ability.
CHECK POINT 5 Being aware of which appliances and activities in your home use the most energy can help you make sound decisions that
allow you to decrease energy consumption and increase savings. The
circle graph in Figure 1.4 shows how energy consumption is distributed throughout a typical home.
Suppose that last year your family spent $2148.72 on natural gas and
electricity. Assuming the circle graph in Figure 1.4 is representative of your family’s energy consumption,
a. Use the appropriate information displayed by the graph to determine a calculation that shows the amount your family spent
on heating and cooling for the year.
b. Use rounding to find a reasonable estimate for this calculation.
Heating and Cooling ,
48%
Water Heater
Lighting, 7%
Computer and Monitor,
2%Clothes Washer and
DryerTV, DVD, VCR, 2%
Dishwasher, 2%
Other
Refrigerator, 6%
12% 11%
10%
The Home Energy Pie
F I G U R E 1 . 4
Source: Natural Home and Garden
20 C H A P T E R 1 Problem Solving and Critical Thinking
60
70
80
90
50
40
30
20
L if
e E
xp ec
ta n
cy
Life Expectancy in the United States, by Year of Birth
199019801960 19701950
10
2010 20202000
78.8 71.8
77.5 70.0
73.1 66.6
74.7 67.1
71.1 65.6
81.1 77.1
81.9 76.2
79.3 74.1
Males Females
Birth Year
F I G U R E 1 . 5
Source: National Center for Health Statistics
Bar graphs are convenient for comparing some measurable attribute of various items. The bars may
be either horizontal or vertical, and their heights or
lengths are used to show the amounts of different
items. Figure 1.5 is an example of a typical bar graph. The graph shows life expectancy for American men
and American women born in various years from
1950 through 2020.
EXAMPLE 6 Applying Estimation and Inductive Reasoning to Data in a Bar Graph
Use the data for men in Figure 1.5 to estimate each of the following:
a. a man’s increased life expectancy, rounded to the nearest hundredth of a year, for each subsequent birth year
b. the life expectancy of a man born in 2030.
SOLUTION
a. One way to estimate increased life expectancy for each subsequent birth year is to generalize from the information given for 1950 (male life
expectancy: 65.6 years) and for 2020 (male life expectancy: 77.1 years). The
average yearly increase in life expectancy is the change in life expectancy
from 1950 to 2020 divided by the change in time from 1950 to 2020.
-
77.1 - 65.6 2020 - 1950
L
L 0.16 Use a calculator. See the Technology box below.
For each subsequent birth year, a man’s life expectancy is increasing by
approximately 0.16 year.
Here is the calculator keystroke sequence needed to perform the computation in
Example 6(a).
� ( � 77.1 � - � 65.6 � ) � � , � � ( � 2020 � - � 1950 � ) � Press � = �on a scientific calculator or � ENTER �on a graphing calculator to display the answer. As specified, we round to the nearest hundredth.
L 0.16
TECHNOLOGY
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 21
Line graphs are often used to illustrate trends over time. Some measure of time, such as months or years, frequently appears on
the horizontal axis. Amounts are generally listed on the vertical
axis. Points are drawn to represent the given information. The
graph is formed by connecting the points with line segments.
Figure 1.6 is an example of a typical line graph. The graph shows the average age at which women in the United States
married for the first time from 1890 through 2015. The years are
listed on the horizontal axis, and the ages are listed on the vertical
axis. The symbol on the vertical axis shows that there is a break
in values between 0 and 20. Thus, the first tick mark on the vertical
axis represents an average age of 20.
Figure 1.6 shows how to find the average age at which women married for the first time in 1980.
Step 1 Locate 1980 on the horizontal axis.
Step 2 Locate the point on the line graph above 1980.
Step 3 Read across to the corresponding age on the vertical axis.
The age is 22. Thus, in 1980, women in the United States married for the first time
at an average age of 22.
b. We can use our computation in part (a) to estimate the life expectancy of an American man born in 2030. The bar graph indicates that men
born in 1950 had a life expectancy of 65.6 years. The year 2030 is 80 years
after 1950, and life expectancy is increasing by approximately 0.16 year
for each subsequent birth year.
= 65.6 + 12.8 = 78.4
L 65.6 + 0.16 * 80
An American man born in 2030 will have a life expectancy of
approximately 78.4 years.
CHECK POINT 6 Use the data for women in Figure 1.5 to estimate each of the following:
a. a woman’s increased life expectancy, rounded to the nearest hundredth of a year, for each subsequent birth year
b. the life expectancy, to the nearest tenth of a year, of a woman born in 2050.
A g e
Women’s Average Age of First Marriage
2000 20151980
Year
19001890 1920 19601940
20
21
22
23
24
25
26
27
28
F I G U R E 1 . 6
Source: U.S. Census Bureau
EXAMPLE 7 Using a Line Graph
The line graph in Figure 1.7 shows the percentage of U.S.
college students who smoked
cigarettes from 1982 through
2014.
a. Find an estimate for the percentage of college
students who smoked
cigarettes in 2010.
Cigarette Use by U.S. College Students
Year
1982 1990 1998 2006 2014
4%
8%
12%
16%
20%
24%
28%
32%
P e rc
e n
t o
f C
o ll
e g e S
tu d
e n
ts
F I G U R E 1 . 7
Source: Rebecca Donatelle, Health The Basics, 10th Edition, Pearson; Monitoring the Future
Study, University of Michigan.
In the calculation at the right, you multiplied before adding. Would it be ok if I performed the operations from left to right and added before multiplying?
No. Arithmetic operations
should be performed in a
specific order. When there are
no grouping symbols, such as
parentheses, multiplication is
always done before addition.
We will have more to say about
the order of operations in
Chapter 5.
GREAT QUESTION!
22 C H A P T E R 1 Problem Solving and Critical Thinking
CHECK POINT 7 Use the line graph in Figure 1.7 at the bottom of the previous page to solve this exercise.
a. Find an estimate for the percentage of college students who smoked cigarettes in 1986.
b. In which four-year period did the percentage of college students who smoked cigarettes increase at the greatest rate?
c. In which years corresponding to a tick mark on the horizontal axis did 24% of college students smoke cigarettes?
d. In which year did the least percentage of college students smoke cigarettes? What percentage of students smoked in that year?
b. In which four-year period did the percentage of college students who smoked cigarettes decrease at the greatest rate?
c. In which year did 30% of college students smoke cigarettes?
SOLUTION
a. Estimating the Percentage Smoking Cigarettes in 2010
Cigarette Use by U.S. College Students
Year
1982 1990 1998 2006 2014
4%
8%
12%
16%
20%
24%
28%
32%
P e rc
e n
t o
f C
o ll
e g e S
tu d
e n
ts
b. Identifying the Period of the Greatest Rate of Decreasing
Cigarette Smoking
Cigarette Use by U.S. College Students
Year
1982 1990 1998 2006 2014
4%
8%
12%
16%
20%
24%
28%
32%
P e rc
e n
t o
f C
o ll
e g e S
tu d
e n
ts
c. Identifying the Year when 30% of College Students Smoked
Cigarettes
Cigarette Use by U.S. College Students
Year
1982 1990 1998 2006 2014
4%
8%
12%
16%
20%
24%
28%
32%
P e rc
e n
t o
f C
o ll
e g e S
tu d
e n
ts
Mathematical Models
We have seen that American men born in 1950 have a life expectancy of
65.6 years, increasing by approximately 0.16 year for each subsequent birth year.
We can use variables to express the life expectancy, E, for American men born x years after 1950.
E = 65.6 + 0.16x
A formula is a statement of equality that uses letters to express a relationship between two or more variables. Thus, E = 65.6 + 0.16x is a formula describing life expectancy, E, for American men born x years after 1950. Be aware that this formula provides estimates of life expectancy, as shown in Table 1.2.
3 Develop mathematical models that estimate relationships between variables.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 23
T A B L E 1 . 2 Comparing Given Data with Estimates Determined by a Formula
Birth Year Life Expectancy: Given
Data Life Expectancy: Formula Estimate
E = 65.6 + 0.16x
1950 65.6 E = 65.6 + 0.16(0) = 65.6 + 0 = 65.6
1960 66.6 E = 65.6 + 0.16(10) = 65.6 + 1.6 = 67.2
1970 67.1 E = 65.6 + 0.16(20) = 65.6 + 3.2 = 68.8
1980 70.0 E = 65.6 + 0.16(30) = 65.6 + 4.8 = 70.4
1990 71.8 E = 65.6 + 0.16(40) = 65.6 + 6.4 = 72.0
2000 74.1 E = 65.6 + 0.16(50) = 65.6 + 8.0 = 73.6
2010 76.2 E = 65.6 + 0.16(60) = 65.6 + 9.6 = 75.2
2020 77.1 E = 65.6 + 0.16(70) = 65.6 + 11.2 = 76.8
x
The process of finding formulas to describe real-world phenomena is called
mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models. We often say that these formulas model, or describe, the relationships among the variables.
The formula in Table 1.2 does not take into account your
current health, lifestyle, and
family history, all of which
could increase or decrease
your life expectancy. Thomas
Perls at Boston University
Medical School, who studies
centenarians, developed a much
more detailed formula for life
expectancy at livingto100.com.
The model takes into account
everything from your stress level
to your sleep habits and gives
you the exact age it predicts you
will live to.
Blitzer Bonus Predicting Your Own Life Expectancy
EXAMPLE 8 Modeling the Cost of Attending a Public College
The bar graph in Figure 1.8 shows the average cost of tuition and fees for public four-year colleges, adjusted for inflation.
a. Estimate the yearly increase in tuition and fees. Round to the nearest dollar.
b. Write a mathematical model that estimates the average cost of tuition and fees, T, at public four-year colleges for the school year ending x years after 2000.
c. Use the mathematical model from part (b) to project the average cost of tuition and fees at public four-year colleges for the school year ending in
2020.
T u
it io
n a
n d
F e e s
Average Cost of Tuition and Fees at Public Four-Year U.S. Colleges
Ending Year in the School Year 2014
8312
2012
7713
2016
9410
2010
6717
2008
5943
2006
5351
2004
4587
2002
3735
2000
3349
$3000 $3500 $4000
$4500
$5000
$5500
$6000
$6500
$7000
$7500
$8000
$8500
$9000
$9500
$10,000
F I G U R E 1 . 8
Source: U.S. Department of Education
24 C H A P T E R 1 Problem Solving and Critical Thinking T
u it
io n
a n
d F
e e s
Average Cost of Tuition and Fees at Public Four-Year U.S. Colleges
Ending Year in the School Year
2 0 1 4
8 3 1 2
2 0 1 2
7 7 1 3
2 0 1 6
9 4 1 0
2 0 1 0
6 7 1 7
2 0 0 8
5 9 4 3
2 0 0 6
5 3 5 1
2 0 0 4
4 5 8 7
2 0 0 2
3 7 3 5
2 0 0 0
3 3 4 9
$3000
$3500
$4000
$4500
$5000
$5500
$6000
$6500
$7000
$7500
$8000
$8500
$9000
$9500
$10,000
F I G U R E 1 . 8 (repeated)
SOLUTION
a. We can use the data in Figure 1.8 from 2000 and 2016 to estimate the yearly increase in tuition and fees.
9410 - 3349 2016 - 2000L
6061
16 = = 378.8125 L 379
Each year the average cost of tuition and fees for public four-year
colleges is increasing by approximately $379.
b. Now we can use variables to obtain a mathematical model that estimates the average cost of tuition and fees, T, for the school year ending x years after 2000.
T = 3349 + 379x
The mathematical model T = 3349 + 379x estimates the average cost of tuition and fees, T, at public four-year colleges for the school year ending x years after 2000.
c. Now let’s use the mathematical model to project the average cost of tuition and fees for the school year ending in 2020. Because 2020 is
20 years after 2000, we substitute 20 for x.
T = 3349 + 379x This is the mathematical model from part (b).
T = 3349 + 379(20) Substitute 20 for x.
= 3349 + 7580 Multiply: 379(20) = 7580.
= 10,929 Add. On a calculator, enter 3349 � + � 379 � : � 20 and press � = � or � ENTER � .
Our model projects that the average cost of tuition and fees at public
four-year colleges for the school year ending in 2020 will be $10,929.
CHECK POINT 8 The bar graph in Figure 1.9 on the next page shows the average cost of tuition and fees for private four-year colleges, adjusted for
inflation.
a. Estimate the yearly increase in tuition and fees. Round to the nearest dollar.
b. Write a mathematical model that estimates the average cost of tuition and fees, T, at private four-year colleges for the school year ending x years after 2000.
c. Use the mathematical model from part (b) to project the average cost of tuition and fees at private four-year colleges for the school year ending
in 2020.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 25
Sometimes a mathematical model gives an estimate that is not a good
approximation or is extended to include values of the variable that do not make
sense. In these cases, we say that model breakdown has occurred. Models that accurately describe data for the past 10 years might not serve as reliable predictions
for what can reasonably be expected to occur in the future. Model breakdown can
occur when formulas are extended too far into the future.
$35,000
$29,000
$25,000
$21,000
$17,000
$31,000
$33,000
$27,000
$23,000
$19,000
T u
it io
n a
n d
F e e s
Average Cost of Tuition and Fees at Private Four-Year U.S. Colleges
Ending Year in the School Year
2002 2004 2006 2008 2010 2012 20142000
15,518
17,272
19,710
21,235
23,712
26,273
29,056
2016
33,480
31,701
$15,000
F I G U R E 1 . 9
Source: U.S. Department of Education
“Questions have intensified about whether going to college
is worthwhile,” says Education Pays, released by the College Board Advocacy & Policy Center. “For the typical student,
the investment pays off very well over the course of a lifetime,
even considering the expense.”
Among the findings in Education Pays:
• Mean (average) full-time earnings with a bachelor’s degree
are approximately $63,000, which is $28,000 more than
high school graduates.
• Compared with a high school graduate, a four-year college
graduate who enrolled in a public university at age 18
will break even by age 33. The college graduate will have
earned enough by then to compensate for being out of the
labor force for four years and for borrowing enough to pay
tuition and fees, shown in Figure 1.8.
Blitzer Bonus Is College Worthwhile?
Fill in each blank so that the resulting statement is true.
1. The process of arriving at an approximate answer to a computation such as 0.79 * 403 is called ____________.
2. A graph that shows how a whole quantity is divided into parts is called a/an _____________.
3. A formula that approximates real-world phenomena is called a/an _____________________.
4. True or False: Decimal numbers are rounded by using the digit to the right of the digit where rounding is to occur.
_______
5. True or False: Line graphs are often used to illustrate trends over time. _______
6. True or False: Mathematical modeling results in formulas that give exact values of real-world phenomena over time.
_______
Concept and Vocabulary Check
26 C H A P T E R 1 Problem Solving and Critical Thinking
Practice Exercises
The bar graph gives the populations of the ten most populous states in the United States. Use the appropriate information displayed by the graph to solve Exercises 1–2.
Population by State of the Ten Most Populace States
California
Texas
Florida
New York
Illinois
Pennsylvania
Ohio
Georgia
North Carolina
Michigan
27,469,114
19,795,791
20,271,272
12,859,995
12,802,503
11,613,423
10,214,860
10,042,802
9,922,576
39,144,818
Source: U.S. Census Bureau
1. Round the population of California to the nearest a. hundred, b. thousand, c. ten-thousand, d. hundred- thousand, e. million, f. ten-million.
2. Select any state other than California. For the state selected, round the population to the nearest a. hundred, b. thousand, c. ten-thousand, d. hundred-thousand, e. million, f. ten million.
Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed?
Martin Gardner
Although most people are familiar with p, the number e is more significant in mathematics, showing up in problems involving population growth and compound interest, and at the heart of the statistical bell curve. One way to think of e is the dollar amount you would have in a savings account at the end of the year if you invested $1 at the beginning of the year and the bank paid an annual interest rate of 100% compounded continuously (compounding interest every trillionth of a second, every quadrillionth of a second, etc.). Although continuous compounding sounds terrific, at the end of the year your $1 would have grown to a mere $e, or $2.72, rounded to the nearest cent. Here is a better approximation for e.
e ≈ 2.718281828459045
In Exercises 3–8, use this approximation to round e as specified.
3. to the nearest thousandth
4. to the nearest ten-thousandth
5. to the nearest hundred-thousandth
6. to the nearest millionth
7. to nine decimal places
8. to ten decimal places
In Exercises 9–34, because different rounding results in different estimates, there is not one single, correct answer to each exercise.
In Exercises 9–22, obtain an estimate for each computation by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer?
9. 359 + 596 10. 248 + 797 11. 8.93 + 1.04 + 19.26 12. 7.92 + 3.06 + 24.36 13. 32.15 - 11.239 14. 46.13 - 15.237 15. 39.67 * 5.5 16. 78.92 * 6.5 17. 0.79 * 414 18. 0.67 * 211 19. 47.83 , 2.9 20. 54.63 , 4.7 21. 32% of 187,253 22. 42% of 291,506
In Exercises 23–34, determine each estimate without using a calculator. Then use a calculator to perform the computation necessary to obtain an exact answer. How reasonable is your estimate when compared to the actual answer?
23. Estimate the total cost of six grocery items if their prices are $3.47, $5.89, $19.98, $2.03, $11.85, and $0.23.
24. Estimate the total cost of six grocery items if their prices are $4.23, $7.79, $28.97, $4.06, $13.43, and $0.74.
25. A full-time employee who works 40 hours per week earns $19.50 per hour. Estimate that person’s annual
income.
26. A full-time employee who works 40 hours per week earns $29.85 per hour. Estimate that person’s annual
income.
27. You lease a car at $605 per month for 3 years. Estimate the total cost of the lease.
28. You lease a car at $415 per month for 4 years. Estimate the total cost of the lease.
29. A raise of $310,000 is evenly distributed among 294 professors. Estimate the amount each professor
receives.
30. A raise of $310,000 is evenly distributed among 196 professors. Estimate the amount each professor
receives.
31. If a person who works 40 hours per week earns $61,500 per year, estimate that person’s hourly wage.
32. If a person who works 40 hours per week earns $38,950 per year, estimate that person’s hourly wage.
33. The average life expectancy in Canada is 80.1 years. Estimate the country’s life expectancy in hours.
34. The average life expectancy in Mozambique is 40.3 years. Estimate the country’s life expectancy in hours.
Practice Plus
In Exercises 35–36, obtain an estimate for each computation without using a calculator. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer?
35. 0.19996 * 107
0.509 36.
0.47996 * 88 0.249
Exercise Set 1.2
37. Ten people ordered calculators. The least expensive was $19.95 and the most expensive was $39.95. Half ordered a
$29.95 calculator. Select the best estimate of the amount
spent on calculators.
a. $240 b. $310 c. $345 d. $355
38. Ten people ordered calculators. The least expensive was $4.95 and the most expensive was $12.95. Half ordered a
$6.95 calculator. Select the best estimate of the amount
spent on calculators.
a. $160 b. $105 c. $75 d. $55
39. Traveling at an average rate of between 60 and 70 miles per hour for 3 to 4 hours, select the best estimate for the distance
traveled.
a. 90 miles b. 190 miles c. 225 miles d. 275 miles
40. Traveling at an average rate of between 40 and 50 miles per hour for 3 to 4 hours, select the best estimate for the distance
traveled.
a. 120 miles b. 160 miles c. 195 miles d. 210 miles
41. Imagine that you counted 60 numbers per minute and continued to count nonstop until you reached 10,000.
Determine a reasonable estimate of the number of hours it
would take you to complete the counting.
42. Imagine that you counted 60 numbers per minute and continued to count nonstop until you reached one million.
Determine a reasonable estimate of the number of days it
would take you to complete the counting.
Application Exercises
The circle graph shows the most important problems for the 16,503,611 high school teenagers in the United States. Use this information to solve Exercises 43–44.
Most Important Problems for High School Teenagers
Drugs, 23%Other,
29%
Social Pressures; Fitting in,
22%
Doing Well in School,
11% Crime and
Violence in School,
4%
Sexual Issues,
4%
Getting into
College, 4%
Getting along with
Parents, 3%
Source: Columbia University
43. Without using a calculator, estimate the number of high school teenagers for whom doing well in school is the most important
problem.
44. Without using a calculator, estimate the number of high school teenagers for whom social pressures and fitting in is
the most important problem.
An online test of English spelling looked at how well people spelled difficult words. The bar graph shows how many people per 100 spelled each word correctly. Use this information to solve Exercises 45–46.
Number of People per 100 Spelling Various Words Correctly
40 10070 80 90605010 30200
inoculate
supersede
accommodation
harass
cemetery
weird
Number of People (per 100)
Source: Vivian Cook, Accomodating Brocolli in the Cemetary or Why Can’t Anybody Spell?, Simon and Schuster, 2004
45. a. Estimate the number of people per 100 who spelled weird correctly.
b. In a group consisting of 8729 randomly selected people, estimate how many more people can correctly spell
weird than inoculate.
46. a. Estimate the number of people per 100 who spelled cemetery correctly.
b. In a group consisting of 7219 randomly selected people, estimate how many more people can correctly spell
cemetery than supersede.
The percentage of U.S. college freshmen claiming no religious affiliation has risen in recent decades. The bar graph shows the percentage of first-year college students claiming no religious affiliation for four selected years from 1980 through 2012. Use this information to solve Exercises 47–48.
Percentage of First-Year U.S. College Students Claiming No Religious Affiliation
Year
Males Females
1980
9.7
6.7
1990
14.0
10.7
2000
16.9
13.2
21.7
2012
26.3
15%
30%
5%
10%
P e rc
e n
ta g e C
la im
in g N
o R
e li
g io
u s
A ffi
li a ti
o n
20%
25%
Source: John Macionis, Sociology, 15th Edition, Pearson, 2014.
47. a. Estimate the average yearly increase in the percentage of first-year college males claiming no religious affiliation.
Round the percentage to the nearest tenth.
b. Estimate the percentage of first-year college males who will claim no religious affiliation in 2020.
48. a. Estimate the average yearly increase in the percentage of first-year college females claiming no religious affiliation.
Round the percentage to the nearest tenth.
b. Estimate the percentage of first-year college females who will claim no religious affiliation in 2020.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 27
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 49–50.
Percent Body Fat in Adults
Age
65 7555453525
20
24
28
32
36
40
P e rc
e n
t B
o d
y F
a t
49. a. Find an estimate for the percent body fat in 45-year-old women.
b. At what age does the percent body fat in women reach a maximum? What is the percent body fat for that
age?
c. At what age do women have 34% body fat?
50. a. Find an estimate for the percent body fat in 25-year-old men.
b. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that
age?
c. At what age do men have 24% body fat?
310
1950
317
1960
326
1970
339
1980
354
1990
369
2000
300
320
340
360
380
400
410
310
330
350
370
390
A v e ra
g e C
a rb
o n
D io
x id
e C
o n
c e n
tr a ti
o n
( p
a rt
s p
e r
m il
li o
n )
390
2010
401
2015
Year
56.98
1950
57.04
1960
57.06
1970
57.35
1980
57.64
1990
57.67
2000
58.44
2015
56.0°
56.6°
57.2°
57.8°
58.4°
59.3°
56.3°
56.9°
57.5°
58.1°
58.7°
59.0°
A v e ra
g e G
lo b
a l
T e m
p e ra
tu re
(d e g re
e s
F a h
re n
h e it
) 58.11
2010
Year
Average Atmospheric Concentration of Carbon Dioxide Average Global Temperature
Source: National Oceanic and Atmospheric Administration
28 C H A P T E R 1 Problem Solving and Critical Thinking
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008.
51. a. Estimate the yearly increase in the average atmospheric concentration of carbon dioxide. Express the answer in
parts per million.
b. Write a mathematical model that estimates the average atmospheric concentration of carbon dioxide, C, in parts per million, x years after 1950.
c. If the trend shown by the data continues, use your mathematical model from part (b) to project the
average atmospheric concentration of carbon dioxide in
2050.
52. a. Estimate the yearly increase in the average global temperature, rounded to the nearest hundredth of a
degree.
b. Write a mathematical model that estimates the average global temperature, T, in degrees Fahrenheit, x years after 1950.
c. If the trend shown by the data continues, use your mathematical model from part (b) to project the average
global temperature in 2050.
Explaining the Concepts
53. What is estimation? When is it helpful to use estimation?
54. Explain how to round 218,543 to the nearest thousand and to the nearest hundred-thousand.
55. Explain how to round 14.26841 to the nearest hundredth and to the nearest thousandth.
56. What does the ≈ symbol mean? 57. In this era of calculators and computers, why is there a need
to develop estimation skills?
58. Describe a circle graph.
59. Describe a bar graph.
There is a strong scientific consensus that human activities are changing the Earth’s climate. Scientists now believe that there is a striking correlation between atmospheric carbon dioxide concentration and global temperature. As both of these variables increase at significant rates, there are warnings of a planetary emergency that threatens to condemn coming generations to a catastrophically diminished future. The bar graphs give the average atmospheric concentration of carbon dioxide and the average global temperature for eight selected years. Use this information to solve Exercises 51–52.
60. Describe a line graph.
61. What does it mean when we say that a formula models real-world phenomena?
62. College students are graduating with the highest debt burden in history. The bar graph shows the mean, or average,
student-loan debt in the United States for six selected
graduating years from 2001 through 2016.
$40,000
$35,000
$25,000
$30,000
$20,000
$15,000
$10,000
M e a n
S tu
d e n
t- L
o a n
D e b
t
Mean Student-Loan Debt in the U.S.
Graduating Year
2001 2013201020072004
33,050
26,682
2016
37,172
23,34922,022
17,562
$5000
Source: Pew Research Center
Describe how to use the data for 2001 and 2016 to estimate
the yearly increase in mean student-loan debt.
63. Explain how to use the estimate from Exercise 62 to write a mathematical model that estimates mean student-loan
debt, D, in dollars, x years after 2001. How can this model be used to project mean student-loan debt in 2020?
64. Describe one way in which you use estimation in a nonacademic area of your life.
65. A forecaster at the National Hurricane Center needs to estimate the time until a hurricane with high probability
of striking South Florida will hit Miami. Is it better to
overestimate or underestimate? Explain your answer.
Critical Thinking Exercises
Make Sense? In Exercises 66–69, determine whether each statement makes sense or does not make sense, and explain your reasoning.
66. When buying several items at the market, I use estimation before going to the cashier to be sure I have enough money
to pay for the purchase.
67. It’s not necessary to use estimation skills when using my calculator.
68. Being able to compute an exact answer requires a different ability than estimating the reasonableness of the
answer.
69. My mathematical model estimates the data for the past 10 years extremely well, so it will serve as an accurate
prediction for what will occur in 2050.
70. Take a moment to read the verse preceding Exercises 3–8 that mentions the numbers p and e, whose decimal representations continue infinitely with no repeating patterns. The verse was
written by the American mathematician (and accomplished
amateur magician!) Martin Gardner (1914–2010), author of
more than 60 books and best known for his “Mathematical
Games” column, which ran in Scientific American for 25 years. Explain the humor in Gardner’s question.
In Exercises 71–74, match the story with the correct graph. The graphs are labeled (a), (b), (c), and (d).
71. As the blizzard got worse, the snow fell harder and harder.
72. The snow fell more and more softly.
73. It snowed hard, but then it stopped. After a short time, the snow started falling softly.
74. It snowed softly, and then it stopped. After a short time, the snow started falling hard.
A m
o u
n t
o f
S n
o w
fa ll
Time
a.
A m
o u
n t
o f
S n
o w
fa ll
Time
b.
A m
o u
n t
o f
S n
o w
fa ll
Time
c.
A m
o u
n t
o f
S n
o w
fa ll
Time
d.
75. American children ages 2 to 17 spend 19 hours 40 minutes per week watching television. (Source: TV-Turnoff Network) From ages 2 through 17, inclusive, estimate the number of
days an American child spends watching television. How
many years, to the nearest tenth of a year, is that?
76. If you spend $1000 each day, estimate how long it will take to spend a billion dollars.
Group Exercises
77. Group members should devise an estimation process that can be used to answer each of the following questions.
Use input from all group members to describe the best
estimation process possible.
a. Is it possible to walk from San Francisco to New York in a year?
b. How much money is spent on ice cream in the United States each year?
78. Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find two graphs
that show “intriguing” data changing from year to year. In
one graph, the data values should be increasing relatively
steadily. In the second graph, the data values should be
decreasing relatively steadily. For each graph selected,
write a mathematical model that estimates the changing
variable x years after the graph’s starting date. Then use each mathematical model to make predictions about what might
occur in the future. Are there circumstances that might
affect the accuracy of the prediction? List some of these
circumstances.
S E C T I O N 1 . 2 Estimation, Graphs, and Mathematical Models 29
30 C H A P T E R 1 Problem Solving and Critical Thinking
1.3 WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Solve problems using the organization of the four-step
problem-solving process.
Problem Solving CRITICAL THINKING AND
problem solving are essential
skills in both school and
work. A model for problem
solving was established by
the charismatic teacher and
mathematician George Polya
(1887–1985) in How to Solve It (Princeton University Press,
Princeton, NJ, 1957). This book,
first published in 1945, has sold
more than one million copies
and is available in 17 languages.
Using a four-step procedure for
problem solving, Polya’s book
demonstrates how to think
clearly in any field.
1 Solve problems using the organization of the four-step problem-solving process.
“If you don’t know where you’re going, you’ll probably end up some place else.” —Yogi Berra
P O LYA’ S F O U R S T E P S I N P R O B L E M S O LV I N G
Step 1 Understand the problem. Read the problem several times. The first reading can serve as an overview. In the second reading, write down what
information is given and determine exactly what it is that the problem requires
you to find.
Step 2 Devise a plan. The plan for solving the problem might involve one or more of these suggested problem-solving strategies:
• Use inductive reasoning to look for a pattern.
• Make a systematic list or a table.
• Use estimation to make an educated guess at the solution. Check the
guess against the problem’s conditions and work backward to eventually
determine the solution.
• Try expressing the problem more simply and solve a similar simpler
problem.
• Use trial and error.
• List the given information in a chart or table.
• Try making a sketch or a diagram to illustrate the problem.
• Relate the problem to a similar problem that you have seen before. Try
applying the procedures used to solve the similar problem to the new one.
• Look for a “catch” if the answer seems too obvious. Perhaps the problem
involves some sort of trick question deliberately intended to lead the
problem solver in the wrong direction.
• Use the given information to eliminate possibilities.
• Use common sense.
Step 3 Carry out the plan and solve the problem.
Step 4 Look back and check the answer. The answer should satisfy the conditions of the problem. The answer should make sense and be reasonable. If
this is not the case, recheck the method and any calculations. Perhaps there is an
alternate way to arrive at a correct solution.
S E C T I O N 1 . 3 Problem Solving 31
Should I memorize Polya’s four steps in problem solving?
Not necessarily. Think of Polya’s four steps as guidelines that will help you organize the
process of problem solving, rather than a list of rigid rules that need to be memorized.
You may be able to solve certain problems without thinking about or using every step in
the four-step process.
GREAT QUESTION!
The very first step in problem solving involves evaluating the given information
in a deliberate manner. Is there enough given to solve the problem? Is the
information relevant to the problem’s solution, or are some facts not necessary to
arrive at a solution?
EXAMPLE 1 Finding What Is Missing
Which necessary piece of information is missing and prevents you from
solving the following problem?
A man purchased five shirts, each at the same discount price. How much
did he pay for them?
SOLUTION
Step 1 Understand the problem. Here’s what is given:
Number of shirts purchased: 5.
We must find how much the man paid for the five shirts.
Step 2 Devise a plan. The amount that the man paid for the five shirts is the number of shirts, 5, times the cost of each shirt. The discount price of each shirt
is not given. This missing piece of information makes it impossible to solve the
problem.
CHECK POINT 1 Which necessary piece of information is missing and prevents you from solving the following problem?
The bill for your meal totaled $20.36, including the tax. How much change
should you receive from the cashier?
EXAMPLE 2 Finding What Is Unnecessary
In the following problem, one more piece of information is given than
is necessary for solving the problem. Identify this unnecessary piece of
information. Then solve the problem.
A roll of E-Z Wipe paper towels contains 100 sheets and costs $1.38.
A comparable brand, Kwik-Clean, contains five dozen sheets per roll
and costs $1.23. If you need three rolls of paper towels, which brand is
the better value?
SOLUTION
Step 1 Understand the problem. Here’s what is given:
E-Z Wipe: 100 sheets per roll; $1.38
Kwik-Clean: 5 dozen sheets per roll; $1.23
Needed: 3 rolls.
We must determine which brand offers the better value.
32 C H A P T E R 1 Problem Solving and Critical Thinking
Step 2 Devise a plan. The brand with the better value is the one that has the lower price per sheet. Thus, we can compare the two brands by finding the cost
for one sheet of E-Z Wipe and one sheet of Kwik-Clean. The price per sheet, or
the unit price, is the price of a roll divided by the number of sheets in the roll. The fact that three rolls are required is not relevant to the problem. This unnecessary
piece of information is not needed to find which brand is the better value.
Step 3 Carry out the plan and solve the problem.
= ×
price of a roll number of sheets per roll
price per sheet =
$1.38 100 sheets
=
E-Z Wipe:
price of a roll number of sheets per roll
price per sheet = Kwik-Clean:
= $0.0138 L $0.01
$1.23 60 sheets
= = $0.0205 L $0.02
By comparing unit prices, we see that E-Z Wipe, at approximately $0.01 per
sheet, is the better value.
Step 4 Look back and check the answer. We can double-check the arithmetic in each of our unit-price computations. We can also see if these unit prices
satisfy the problem’s conditions. The product of each brand’s price per sheet
and the number of sheets per roll should result in the given price for a roll.
E-Z Wipe: Check $0.0138
$0.0138 * 100 = $1.38 Kwik-Clean: Check $0.0205
$0.0205 * 60 = $1.23
The unit prices satisfy the problem’s conditions.
A generalization of our work in Example 2 allows you to compare
different brands and make a choice among various products of different sizes.
When shopping at the supermarket, a useful number to keep in mind is a
product’s unit price. The unit price is the total price divided by the total units. Among comparable brands, the best value is the product with the lowest unit
price, assuming that the units are kept uniform.
The word per is used to state unit prices. For example, if a 12-ounce box of cereal sells for $3.00, its unit price is determined as follows:
Unit price = total price
total units =
+3.00 12 ounces
= +0.25 per ounce.
CHECK POINT 2 Solve the following problem. If the problem contains information that is not relevant to its solution, identify this unnecessary piece
of information.
A manufacturer packages its apple juice in bottles and boxes. A 128-ounce
bottle costs $5.39, and a 9-pack of 6.75-ounce boxes costs $3.15. Which
packaging option is the better value?
In 200% of Nothing (John Wiley & Sons, 1993), author
A. K. Dewdney writes, “It must
be something of a corporate
dream come true when a
company charges more for a
product and no one notices.”
He gives two examples of
“sneaky pricejacks,” both
easily detected using unit
prices. The manufacturers of
Mennen Speed Stick deodorant
increased the size of the
package that held the stick, left
the price the same, and reduced
the amount of actual deodorant
in the stick from 2.5 ounces to
2.25. Fabergé’s Brut left the
price and size of its cologne
jar the same, but reduced its
contents from 5 ounces to 4.
Surprisingly, the new jar read,
“Now, more Brut!” Consumer Reports contacted Fabergé to see how this could be possible.
Their response: The new jar
contained “more fragrance.”
Consumer Reports moaned, “Et tu Brut?”
Blitzer Bonus Unit Prices and Sneaky Pricejacks
S E C T I O N 1 . 3 Problem Solving 33
EXAMPLE 3 Applying the Four-Step Procedure
By paying $100 cash up front and the balance at $20 a week, how long will it
take to pay for a bicycle costing $680?
SOLUTION
Step 1 Understand the problem. Here’s what is given:
Cost of the bicycle: $680
Amount paid in cash: $100
Weekly payments: $20.
If necessary, consult a dictionary to look up any unfamiliar words. The word
balance means the amount still to be paid. We must find the balance to determine the number of weeks required to pay off the bicycle.
Step 2 Devise a plan. Subtract the amount paid in cash from the cost of the bicycle. This results in the amount still to be paid. Because weekly payments
are $20, divide the amount still to be paid by 20. This will give the number of
weeks required to pay for the bicycle.
Step 3 Carry out the plan and solve the problem. Begin by finding the balance, the amount still to be paid for the bicycle.
+680 -+100 +580
Now divide the $580 balance by $20, the payment per week. The result of the
division is the number of weeks needed to pay off the bicycle.
+580 +20
week
= +580 * week
+20 =
580 weeks
20 = 29 weeks
It will take 29 weeks to pay for the bicycle.
Step 4 Look back and check the answer. We can certainly double-check the arithmetic either by hand or with a calculator. We can also see if the answer,
29 weeks to pay for the bicycle, satisfies the condition that the bicycle costs
$680.
$20
* 29 $580
+580 ++100 +680
The answer of 29 weeks satisfies the condition that the cost of the bicycle
is $680.
Is there a strategy I can use to determine whether I understand a problem?
An effective way to see if you
understand a problem is to
restate the problem in your
own words.
“A problem well stated is a
problem half solved.”
—Charles Franklin Kettering
GREAT QUESTION!
cost of the bicycle
amount paid in cash
amount still to be paid
total of weekly payments amount paid in cash
cost of bicycle
weekly payment
number of weeks
total of weekly payments
CHECK POINT 3 By paying $350 cash up front and the balance at $45 per month, how long will it take to pay for a computer costing $980?
Making lists is a useful strategy in problem solving.
34 C H A P T E R 1 Problem Solving and Critical Thinking
SOLUTION
Step 1 Understand the problem. The total change must always be 50 cents. One possible coin combination is two quarters. Another is five dimes. We need
to count all such combinations.
Step 2 Devise a plan. Make a list of all possible coin combinations. Begin with the coins of larger value and work toward the coins of smaller value.
Step 3 Carry out the plan and solve the problem. First we must find all of the coins that are not pennies but can combine to form 50 cents. This includes
half-dollars, quarters, dimes, and nickels. Now we can set up a table. We will
use these coins as table headings.
Half-Dollars Quarters Dimes Nickels
Each row in the table will represent one possible combination for exact
change. We start with the largest coin, the half-dollar. Only one half-dollar is
needed to make exact change. No other coins are needed. Thus, we put a 1
in the half-dollars column and 0s in the other columns to represent the first
possible combination.
Half-Dollars Quarters Dimes Nickels
1 0 0 0
Likewise, two quarters are also exact change for 50 cents. We put a 0 in the
half-dollars column, a 2 in the quarters column, and 0s in the columns for
dimes and nickels.
Half-Dollars Quarters Dimes Nickels
1 0 0 0
0 2 0 0
In this manner, we can find all possible combinations for exact change for the
50-cent toll. These combinations are shown in Table 1.3.
T A B L E 1 . 3 Exact Change for 50 Cents: No Pennies
Half-Dollars Quarters Dimes Nickels
1 0 0 0
0 2 0 0
0 1 2 1
0 1 1 3
0 1 0 5
0 0 5 0
0 0 4 2
0 0 3 4
0 0 2 6
0 0 1 8
0 0 0 10
Think about the following
questions carefully before
answering because each
contains some sort of trick or
catch.
Sample: Do they have a fourth
of July in England?
Answer: Of course they do.
However, there is no national
holiday on that date!
See if you can answer the
questions that follow without
developing mental whiplash.
The answers appear in the
answer section.
1. A farmer had 17 sheep. All but 12 died. How many
sheep does the farmer have
left?
2. Some months have 30 days. Some have 31. How many
months have 28 days?
3. A doctor had a brother, but this brother had no
brothers. What was the
relationship between doctor
and brother?
4. If you had only one match and entered a log cabin
in which there was a
candle, a fireplace, and a
woodburning stove, which
should you light first?
Blitzer Bonus Trick Questions
EXAMPLE 4 Solving a Problem by Making a List
Suppose you are an engineer programming the automatic gate for a 50-cent
toll. The gate should accept exact change only. It should not accept pennies.
How many coin combinations must you program the gate to accept?
S E C T I O N 1 . 3 Problem Solving 35
Count the coin combinations shown in Table 1.3. How many coin combinations must the gate accept? You must program the gate to accept
11 coin combinations.
Step 4 Look back and check the answer. Double-check Table 1.3 to make sure that no possible combinations have been omitted and that the
total in each row is 50 cents. Double-check your count of the number of
combinations.
CHECK POINT 4 Suppose you are an engineer programming the automatic gate for a 30-cent toll. The gate should accept exact change only. It should not
accept pennies. How many coin combinations must you program the gate to
accept?
Sketches and diagrams are sometimes useful in problem solving.
EXAMPLE 5 Solving a Problem by Using a Diagram
Four runners are in a one-mile race: Maria, Aretha, Thelma, and Debbie.
Points are awarded only to the women finishing first or second. The first-place
winner gets more points than the second-place winner. How many different
arrangements of first- and second-place winners are possible?
SOLUTION
Step 1 Understand the problem. Three possibilities for first and second position are
Maria-Aretha
Maria-Thelma
Aretha-Maria.
Notice that Maria finishing first and Aretha finishing second is a different
outcome than Aretha finishing first and Maria finishing second. Order
makes a difference because the first-place winner gets more points than the
second-place winner. We must count all possibilities for first and second
position.
Step 2 Devise a plan. If Maria finishes first, then each of the other three runners could finish second:
Aretha Maria Thelma
Debbie
Maria-Aretha Maria-Thelma Maria-Debbie
First place Second place Possibilities for first
and second place
Similarly, we can list each woman as the possible first-place runner. Then we
will list the other three women as possible second-place runners. Next we will
determine the possibilities for first and second place. This diagram will show
how the runners can finish first or second.
36 C H A P T E R 1 Problem Solving and Critical Thinking
Step 4 Look back and check the answer. Check the diagram in Figure 1.10 to make sure that no possible first- and second-place outcomes have been left
out. Double-check your count for the winning pairs of runners.
Aretha
Maria Thelma
Debbie
Maria-Aretha
Maria-Thelma
Maria-Debbie
First place Second place Possibilities for first
and second place
Maria
Aretha Thelma
Debbie
Aretha-Maria
Aretha-Thelma
Aretha-Debbie
Maria
Thelma Aretha
Debbie
Thelma-Maria
Thelma-Aretha
Thelma-Debbie
Maria
Debbie Aretha
Thelma
Debbie-Maria
Debbie-Aretha
Debbie-Thelma
F I G U R E 1 . 1 0 Possible ways for four runners to finish first and second
CHECK POINT 5 Your “lecture wardrobe” is rather limited—just two pairs of jeans to choose from (one blue, one black) and three T-shirts to choose from
(one beige, one yellow, and one blue). How many different outfits can you
form?
B
C
D
A
E
128
195
115 147
145194
169
114
180
116
$
F I G U R E 1 . 1 1
EXAMPLE 6 Using a Reasonable Option to Solve a Problem with More Than One Solution
A sales director who lives in city A is required to fly to regional offices in cities B, C, D, and E. Other than starting and ending the trip in city A, there are no restrictions as to the order in which the other four cities are visited.
The one-way fares between each of the cities are given in Table 1.4. A diagram that illustrates this information is shown in Figure 1.11.
T A B L E 1 . 4 One-Way Airfares
A B C D E
A * $180 $114 $147 $128
B $180 * $116 $145 $195
C $114 $116 * $169 $115
D $147 $145 $169 * $194
E $128 $195 $115 $194 *
Give the sales director an order for visiting cities B, C, D, and E once, returning home to city A, for less than $750.
In Chapter 14, we will be studying diagrams, called graphs, that provide structures for describing relationships. In Example 6, we use such a diagram to
illustrate the relationship between cities and one-way airfares between them.
Step 3 Carry out the plan and solve the problem. Now we complete the diagram started in step 2. The diagram is shown in Figure 1.10.
Count the number of possibilities shown under the third column, “Possibilities
for first and second place.” Can you see that there are 12 possibilities?
Therefore, 12 different arrangements of first- and second-place winners are
possible.
S E C T I O N 1 . 3 Problem Solving 37
SOLUTION
Step 1 Understand the problem. There are many ways to visit cities B, C, D, and E once, and return home to A. One route is
A, E, D, C, B, A. A E D C
B A
The cost of this trip involves the sum of five costs, shown in both Table 1.4 and Figure 1.11:
$128 + $194 + $169 + $116 + $180 = $787.
We must find a route that costs less than $750.
Step 2 Devise a plan. The sales director starts at city A. From there, fly to the city to which the airfare is cheapest. Then from there fly to the next city
to which the airfare is cheapest, and so on. From the last of the cities, fly home
to city A. Compute the cost of this trip to see if it is less than $750. If it is not, use trial and error to find other possible routes and select an order (if there is
one) whose cost is less than $750.
Step 3 Carry out the plan and solve the problem. See Figure 1.12. The route is indicated using red lines with arrows.
• Start at A.
• Choose the line segment with the smallest number: 114. Fly from A to C. (cost: $114)
• From C, choose the line segment with the smallest number that does not lead to A: 115. Fly from C to E. (cost: $115)
• From E, choose the line segment with the smallest number that does not lead to a city already visited: 194. Fly from E to D. (cost: $194)
• From D, there is little choice but to fly to B, the only city not yet visited. (cost: $145)
• From B, return home to A. (cost: $180)
The route that we are considering is
A, C, E, D, B, A.
Let’s see if the cost is less than $750. The cost is
$114 + $115 + $194 + $145 + $180 = $748.
Because the cost is less than $750, the sales director can follow the order
A, C, E, D, B, A.
Step 4 Look back and check the answer. Use Table 1.4 on the previous page or Figure 1.12 to verify that the five numbers used in the sum shown above are correct. Use estimation to verify that $748 is a reasonable cost for
the trip.
CHECK POINT 6 As in Example 6, a sales director who lives in city A is required to fly to regional offices in cities B, C, D, and E. The diagram in Figure 1.13 shows the one-way airfares between any two cities. Give the sales director an order for visiting cities B, C, D, and E once, returning home to city A, for less than $1460.
B
C
D
A
E
128
195
115 147
145194
169
114
180
116
F I G U R E 1 . 1 2
A
205
302 305
500
200185
360165
340
320 C
B
D
E
F I G U R E 1 . 1 3
38 C H A P T E R 1 Problem Solving and Critical Thinking
Fill in each blank so that the resulting statement is true.
1. The first step in problem solving is to read the problem several times in order to _____________ the problem.
2. The second step in problem solving is to ______________ for solving the problem.
3. True or False: Polya’s four steps in problem solving make it possible to obtain answers to problems even if necessary
pieces of information are missing. _______
4. True or False: When making a choice between various sizes of a product, the best value is the size with the lowest price.
_______
Concept and Vocabulary Check
Everyone can become a better, more confident problem solver. As in learning any other skill, learning problem solving requires hard work and patience. Work as many problems as possible in this Exercise Set. You may feel confused once in a while, but do not be discouraged. Thinking about a particular problem and trying different methods can eventually lead to new insights. Be sure to check over each answer carefully!
Practice and Application Exercises
In Exercises 1–4, what necessary piece of information is missing that prevents solving the problem?
1. If a student saves $35 per week, how long will it take to save enough money to buy a computer?
2. If a steak sells for $8.15, what is the cost per pound?
3. If it takes you 4 minutes to read a page in a book, how many words can you read in one minute?
4. By paying $1500 cash and the balance in equal monthly payments, how many months would it take to pay for a car
costing $12,495?
In Exercises 5–8, one more piece of information is given than is necessary for solving the problem. Identify this unnecessary piece of information. Then solve the problem.
5. A salesperson receives a weekly salary of $350. In addition, $15 is paid for every item sold in excess of 200
items. How much extra is received from the sale of 212
items?
6. You have $250 to spend and you need to purchase four new tires. If each tire weighs 21 pounds and costs $42 plus $2.50
tax, how much money will you have left after buying the
tires?
7. A parking garage charges $2.50 for the first hour and $0.50 for each additional hour. If a customer gave the parking
attendant $20.00 for parking from 10 a.m. to 3 p.m., how
much did the garage charge?
8. An architect is designing a house. The scale on the plan is 1 inch = 6 feet. If the house is to have a length of 90 feet and a width of 30 feet, how long will the line representing
the house’s length be on the blueprint?
Use Polya’s four-step method in problem solving to solve Exercises 9–44.
9. a. Which is the better value: a 15.3-ounce box of cereal for $3.37 or a 24-ounce box of cereal for $4.59?
b. The supermarket displays the unit price for the 15.3-ounce box in terms of cost per ounce, but displays
the unit price for the 24-ounce box in terms of cost per
pound. What are the unit prices, to the nearest cent,
given by the supermarket?
c. Based on your work in parts (a) and (b), does the better value always have the lower displayed unit price?
Explain your answer.
10. a. Which is the better value: a 12-ounce jar of honey for $2.25 or an 18-ounce jar of honey for $3.24?
b. The supermarket displays the unit price for the 12-ounce jar in terms of cost per ounce, but displays the unit price
for the 18-ounce jar in terms of cost per quart. Assuming
32 ounces in a quart, what are the unit prices, to the
nearest cent, given by the supermarket?
c. Based on your work in parts (a) and (b), does the better value always have the lower displayed unit price?
Explain your answer.
11. One person earns $48,000 per year. Another earns $3750 per month. How much more does the first person earn in a
year than the second?
12. At the beginning of a year, the odometer on a car read 25,124 miles. At the end of the year, it read 37,364 miles. If
the car averaged 24 miles per gallon, how many gallons of
gasoline did it use during the year?
Use the following movie-rental options to solve Exercises 13–14.
Redbox
• Rent DVDs from vending machines: $1.00 per DVD per night
iTunes
• New films (watching online): $3.99/24 hours
• Other films (watching online): $2.99/24 hours
Netflix
• Unlimited streaming (watching online): $7.99/month
• One DVD at a time by mail: $7.99/month
Exercise Set 1.3
S E C T I O N 1 . 3 Problem Solving 39
13. In one month, you rent seven DVDs from a Redbox machine. You return four of the movies after one night, but
keep the other three for two nights. Would you have spent
more or less on Netflix’s unlimited streaming option? How
much more or less?
14. Suppose that you have the Netflix unlimited streaming plan. Because iTunes has two new films that are not available
on Netflix, you download the movies on iTunes, each for
24 hours. What is your total movie-rental cost for the
month?
Acetaminophen is in many non-prescription medications, making it easy to get more than the 4000 milligrams per day linked to liver damage and the recommended 3250-milligram daily maximum. Tylenol Extra Strength contains 500 milligrams of acetaminophen per pill. NyQuil Cold and Flu contains 325 milligrams of acetaminophen per pill. Use this information to solve Exercises 15–16.
15. a. What is the maximum number of Tylenol Extra Strength pills that should be taken in 24 hours?
b. If you take one Tylenol Extra Strength pill per hour for three hours, what is the maximum number of NyQuil
Cold and Flu pills that should be taken for the remainder
of 24 hours?
16. a. What is the maximum number of NyQuil Cold and Flu pills that should be taken should be taken in
24 hours?
b. If you take one Tylenol Extra Strength pill per hour for four hours, what is the maximum number of NyQuil
Cold and Flu pills that should be taken for the remainder
of 24 hours?
17. A television sells for $750. Instead of paying the total amount at the time of the purchase, the same television
can be bought by paying $100 down and $50 a month for
14 months. How much is saved by paying the total amount at
the time of the purchase?
18. In a basketball game, the Bulldogs scored 34 field goals, each counting 2 points, and 13 foul goals, each counting 1 point.
The Panthers scored 38 field goals and 8 foul goals. Which
team won? By how many points did it win?
19. Calculators were purchased at $65 per dozen and sold at $20 for three calculators. Find the profit on six dozen
calculators.
20. Pens are bought at $0.95 per dozen and sold in groups of four for $2.25. Find the profit on 15 dozen pens.
21. Each day a small business owner sells 200 pizza slices at $1.50 per slice and 85 sandwiches at $2.50 each. If business
expenses come to $60 per day, what is the owner’s profit for
a 10-day period?
22. A college tutoring center pays math tutors $8.15 per hour. Tutors earn an additional $2.20 per hour for each hour over
40 hours per week. A math tutor worked 42 hours one week
and 45 hours the second week. How much did the tutor earn
in this two-week period?
23. A car rents for $220 per week plus $0.25 per mile. Find the rental cost for a two-week trip of 500 miles for a group of
three people.
24. A college graduate receives a salary of $2750 a month for her first job. During the year she plans to spend $4800 for
rent, $8200 for food, $3750 for clothing, $4250 for household
expenses, and $3000 for other expenses. With the money
that is left, she expects to buy as many shares of stock at
$375 per share as possible. How many shares will she be able
to buy?
25. Charlene decided to ride her bike from her home to visit her friend Danny. Three miles away from home, her bike got
a flat tire and she had to walk the remaining two miles to
Danny’s home. She could not repair the tire and had to walk
all the way back home. How many more miles did Charlene
walk than she rode?
26. A store received 200 containers of juice to be sold by April 1. Each container cost the store $0.75 and sold for $1.25. The
store signed a contract with the manufacturer in which the
manufacturer agreed to a $0.50 refund for every container
not sold by April 1. If 150 containers were sold by April 1,
how much profit did the store make?
27. A storeowner ordered 25 calculators that cost $30 each. The storeowner can sell each calculator for $35. The
storeowner sold 22 calculators to customers. He had to
return 3 calculators and pay a $2 charge for each returned
calculator. Find the storeowner’s profit.
28. New York City and Washington, D.C. are about 240 miles apart. A car leaves New York City at noon traveling directly
south toward Washington, D.C. at 55 miles per hour. At the
same time and along the same route, a second car leaves
Washington, D.C. bound for New York City traveling directly
north at 45 miles per hour. How far has each car traveled
when the drivers meet for lunch at 2:24 p.m.?
29. An automobile purchased for $23,000 is worth $2700 after 7 years. Assuming that the car’s value depreciated steadily
from year to year, what was it worth at the end of the third
year?
30. An automobile purchased for $34,800 is worth $8550 after 7 years. Assuming that the car’s value depreciated steadily
from year to year, what was it worth at the end of the third
year?
31. A vending machine accepts nickels, dimes, and quarters. Exact change is needed to make a purchase. How many ways
can a person with five nickels, three dimes, and two quarters
make a 45-cent purchase from the machine?
32. How many ways can you make change for a quarter using only pennies, nickels, and dimes?
33. The members of the Student Activity Council on your campus are meeting to select two speakers for a month-long
event celebrating artists and entertainers. The choices are
Emma Watson, George Clooney, Leonardo DiCaprio, and
Jennifer Lawrence. How many different ways can the two
speakers be selected?
34. The members of the Student Activity Council on your campus are meeting to select two speakers for a month-long
event exploring why some people are most likely to succeed.
The choices are Bill Gates, Oprah Winfrey, Mark Zuckerberg,
Hillary Clinton, and Steph Curry. How many different ways
can the two speakers be selected?
40 C H A P T E R 1 Problem Solving and Critical Thinking
35. If you spend $4.79, in how many ways can you receive change from a five-dollar bill?
36. If you spend $9.74, in how many ways can you receive change from a ten-dollar bill?
37. You throw three darts at the board shown. Each dart hits the board and scores a 1, 5, or 10. How many different total
scores can you make?
38. Suppose that you throw four darts at the board shown. With these four darts, there are 16 ways to hit four different
numbers whose sum is 100. Describe one way you can hit
four different numbers on the board that total 100.
39. Five housemates (A, B, C, D, and E) agreed to share the expenses of a party equally. If A spent $42, B spent $10,
C spent $26, D spent $32, and E spent $30, who owes money
after the party and how much do they owe? To whom is
money owed, and how much should they receive? In order
to resolve these discrepancies, who should pay how much to
whom?
40. Six houses are spaced equally around a circular road. If it takes 10 minutes to walk from the first house to the third
house, how long would it take to walk all the way around the
road?
41. If a test has four true/false questions, in how many ways can there be three answers that are false and one answer that is
true?
42. There are five people in a room. Each person shakes the hand of every other person exactly once. How many
handshakes are exchanged?
43. Five runners, Andy, Beth, Caleb, Darnell, and Ella, are in a one-mile race. Andy finished the race 7 seconds before
Caleb. Caleb finished the race 2 seconds before Beth. Beth
finished the race 6 seconds after Darnell. Ella finished the
race 8 seconds after Darnell. In which order did the runners
finish the race?
44. Eight teams are competing in a volleyball tournament. Any team that loses a game is eliminated from the tournament.
How many games must be played to determine the
tournament winner?
In Exercises 45–46, you have three errands to run around town, although in no particular order. You plan to start and end at home. You must go to the bank, the post office, and the dry cleaners. Distances, in miles, between any two of these locations are given in the diagram.
Bank
Home
Post Office
Dry Cleaners
1.5 3.5
43
5
1
45. Determine a route whose distance is less than 12 miles for running the errands and returning home.
46. Determine a route whose distance exceeds 12 miles for running the errands and returning home.
47. The map shows five western states. Trace a route on the map that crosses each common state border exactly once.
WY
UT CO
AZ NM
48. The layout of a city with land masses and bridges is shown. Trace a route that shows people how to walk through the
city so as to cross each bridge exactly once.
South Bank
North Bank
River
49. Jose, Bob, and Tony are college students living in adjacent dorm rooms. Bob lives in the middle dorm room. Their
majors are business, psychology, and biology, although not
necessarily in that order. The business major frequently uses
the new computer in Bob’s dorm room when Bob is in class.
The psychology major and Jose both have 8 a.m. classes, and
the psychology major knocks on Jose’s wall to make sure he
is awake. Determine Bob’s major.
50. The figure represents a map of 13 countries. If countries that share a common border cannot be the same color, what is the
minimum number of colors needed to color the map?
S E C T I O N 1 . 3 Problem Solving 41
The sudoku (pronounced: sue-DOE-koo) craze, a number puzzle popular in Japan, hit the United States in 2005. A sudoku (“single number”) puzzle consists of a 9-by-9 grid of 81 boxes subdivided into nine 3-by-3 squares. Some of the square boxes contain numbers. Here is an example:
The objective is to fill in the remaining squares so that every row, every column, and every 3-by-3 square contains each of the digits from 1 through 9 exactly once. (You can work this puzzle in Exercise 70, perhaps consulting one of the dozens of sudoku books in which the numerals 1 through 9 have created a cottage industry for publishers. There’s even a Sudoku for Dummies.)
Trying to slot numbers into small checkerboard grids is not unique to sudoku. In Exercises 51–54, we explore some of the intricate patterns in other arrays of numbers, including magic squares. A magic square is a square array of numbers arranged so that the numbers in all rows, all columns, and the two diagonals have the same sum. Here is an example of a magic square in which the sum of the numbers in each row, each column, and each diagonal is 15:
+ + = + + = + + =
+ + =
+ + =
+ + =
+ + =+ + =
8
4
6 53 1
9 7 2
Exercises 51–52 are based on magic squares. (Be sure you have read the preceding discussion.)
51. a. Use the properties of a magic square to fill in the missing numbers.
5 18 15
25
b. Show that the number of letters in the word for each number in the square in part (a) generates another
magic square.
52. a. Use the properties of a magic square to fill in the missing numbers.
96 37 45
57
43
25
788223
b. Show that if you reverse the digits for each number in the square in part (a), another magic square is generated.
(Source for the alphamagic square in Exercise 51 and the mirrormagic square in Exercise 52: Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005)
53. As in sudoku, fill in the missing numbers in the 3-by-3 square so that it contains each of the digits from 1 through 9 exactly
once. Furthermore, in this antimagic square, the rows, the columns, and the two diagonals must have different sums.
9
3
7 1
5 54. The missing numbers in the 4-by-4 array are one-digit
numbers. The sums for each row, each column, and one
diagonal are listed in the voice balloons outside the array.
Find the missing numbers.
3 6
44
81
29
55. Some numbers in the printing of a division problem have become illegible. They are designated below by *. Fill in the blanks.
1**
**)4*** 28
*56
***
***
***
0
Explaining the Concepts
In Exercises 56–58, explain the plan needed to solve the problem.
56. If you know how much was paid for several pounds of steak, find the cost of one pound.
42 C H A P T E R 1 Problem Solving and Critical Thinking
57. If you know a person’s age, find the year in which that person was born.
58. If you know how much you earn each hour, find your yearly income.
59. Write your own problem that can be solved using the four-step procedure. Then use the four steps to solve the
problem.
Critical Thinking Exercises
Make Sense? In Exercises 60–63, determine whether each statement makes sense or does not make sense, and explain your reasoning.
60. Polya’s four steps in problem solving make it possible for me to solve any mathematical problem easily and quickly.
61. I used Polya’s four steps in problem solving to deal with a personal problem in need of a creative solution.
62. I find it helpful to begin the problem-solving process by restating the problem in my own words.
63. When I get bogged down with a problem, there’s no limit to the amount of time I should spend trying to solve it.
64. Gym lockers are to be numbered from 1 through 99 using metal numbers to be nailed onto each locker. How many 7s
are needed?
65. You are on vacation in an isolated town. Everyone in the town was born there and has never left. You develop a
toothache and check out the two dentists in town. One
dentist has gorgeous teeth and one has teeth that show
the effects of poor dental work. Which dentist should you
choose and why?
66. India Jones is standing on a large rock in the middle of a square pool filled with hungry, man-eating piranhas. The
edge of the pool is 20 feet away from the rock. India’s mom
wants to rescue her son, but she is standing on the edge of
the pool with only two planks, each 19 1 2 feet long. How can
India be rescued using the two planks?
67. One person tells the truth on Monday, Tuesday, Wednesday, and Thursday, but lies on all other days. A second person lies
on Tuesday, Wednesday, and Thursday, but tells the truth on
all other days. If both people state “I lied yesterday,” then
what day of the week is it today?
68. (This logic problem dates back to the eighth century.) A farmer needs to take his goat, wolf, and cabbage across a
stream. His boat can hold him and one other passenger (the
goat, wolf, or cabbage). If he takes the wolf with him, the
goat will eat the cabbage. If he takes the cabbage, the wolf
will eat the goat. Only when the farmer is present are the
cabbage and goat safe from their respective predators. How
does the farmer get everything across the stream?
69. As in sudoku, fill in the missing numbers along the sides of the
triangle so that it contains each of
the digits from 1 through 9 exactly
once. Furthermore, each side of the
triangle should contain four digits
whose sum is 17.
70. Solve the sudoku puzzle in the top of the left column on page 41.
71. A version of this problem, called the missing dollar problem, first appeared in 1933. Three people eat at a restaurant and
receive a total bill for $30. They divide the amount equally
and pay $10 each. The waiter gives the bill and the $30 to
the manager, who realizes there is an error: The correct
charge should be only $25. The manager gives the waiter
five $1 bills to return to the customers, with the restaurant’s
apologies. However, the waiter is dishonest, keeping $2 and
giving back only $3 to the customers. In conclusion, each of
the three customers has paid $9 and the waiter has stolen
$2, giving a total of $29. However, the original bill was $30.
Where has the missing dollar gone?
72. A firefighter spraying water on a fire stood on the middle rung of a ladder. When the smoke became less thick, the
firefighter moved up 4 rungs. However it got too hot, so
the firefighter backed down 6 rungs. Later, the firefighter
went up 7 rungs and stayed until the fire was out. Then, the
firefighter climbed the remaining 4 rungs and entered the
building. How many rungs does the ladder have?
73. The Republic of Margaritaville is composed of four states: A, B, C, and D. According to the country’s constitution, the
congress will have 30 seats, divided among the four states
according to their respective populations. The table shows
each state’s population.
POPULATION OF MARGARITAVILLE BY STATE
State A B C D Total
Population
(in thousands) 275 383 465 767 1890
Allocate the 30 congressional seats among the four states in
a fair manner.
Group Exercises
Exercises 74–78 describe problems that have many plans for finding an answer. Group members should describe how the four steps in problem solving can be applied to find a solution. It is not necessary to actually solve each problem. Your professor will let the group know if the four steps should be described verbally by a group spokesperson or in essay form.
74. How much will it cost to install bicycle racks on campus to encourage students to use bikes, rather than cars, to get to
campus?
75. How many new counselors are needed on campus to prevent students from waiting in long lines for academic advising?
76. By how much would taxes in your state have to be increased to cut tuition at community colleges and state universities in
half?
77. Is your local electric company overcharging its customers?
78. Should solar heating be required for all new construction in your community?
79. Group members should describe a problem in need of a solution. Then, as in Exercises 74–78, describe how the four
steps in problem solving can be applied to find a solution.
1 3
2
Chapter Summary, Review, and Test 43
Review Exercises
Chapter Summary, Review, and Test
SUMMARY – DEFINITIONS AND CONCEPTS EXAMPLES
1.1 Inductive and Deductive Reasoning
a. Inductive reasoning is the process of arriving at a general conclusion based on observations of specific examples. The conclusion is called a conjecture or a hypothesis. A case for which a conjecture is false is
called a counterexample.
Ex. 1, p. 3; Ex. 2, p. 4; Ex. 3, p. 5; Ex. 4, p. 7
b. Deductive reasoning is the process of proving a specific conclusion from one or more general statements. The statement that is proved is called a theorem.
Ex. 5, p. 9
1.2 Estimation, Graphs, and Mathematical Models
a. The procedure for rounding whole numbers is given in the box on page 15. The symbol ≈ means is approximately equal to.
Ex. 1, p. 15
b. Decimal parts of numbers are rounded in nearly the same way as whole numbers. However, digits to the right of the rounding place are dropped.
Ex. 2, p. 16
c. Estimation is the process of arriving at an approximate answer to a question. Computations can be estimated by using rounding that results in simplified arithmetic.
Ex. 3, p. 17; Ex. 4, p. 17
d. Estimation is useful when interpreting information given by circle, bar, or line graphs. Ex. 5, p. 19; Ex. 6, p. 20; Ex. 7, p. 21
e. The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models.
Ex. 8, p. 23
1.3 Problem Solving
Polya’s Four Steps in Problem Solving
1. Understand the problem.
2. Devise a plan.
3. Carry out the plan and solve the problem.
4. Look back and check the answer.
Ex. 1, p. 31; Ex. 2, p. 31; Ex. 3, p. 33; Ex. 4, p. 34; Ex. 5, p. 35; Ex. 6, p. 36
1.1
1. Which reasoning process is shown in the following example? Explain your answer.
All books by Stephen King have made the best-seller list.
Carrie is a novel by Stephen King. Therefore, Carrie was on the best-seller list.
2. Which reasoning process is shown in the following example? Explain your answer.
All books by Stephen King have made the best-seller
list. Therefore, it is highly probable that the novel
King is currently working on will make the best-seller
list.
In Exercises 3–10, identify a pattern in each list of numbers. Then use this pattern to find the next number.
3. 4, 9, 14, 19, ______ 4. 7, 14, 28, 56, _______
5. 1, 3, 6, 10, 15, ______ 6. 3
4 ,
3
5 ,
1
2 ,
3
7 , _____
7. 40, - 20, 10, - 5, ____ 8. 40, - 20, - 80, - 140, ________ 9. 2, 2, 4, 6, 10, 16, 26, ______
10. 2, 6, 12, 36, 72, 216, _______
11. Identify a pattern in the following sequence of figures. Then use the pattern to find the next figure in the sequence.
, , , ,
In Exercises 12–13, use inductive reasoning to predict the next line in each sequence of computations. Then perform the arithmetic to determine whether your conjecture is correct.
12. 2 = 4 - 2 2 + 4 = 8 - 2
2 + 4 + 8 = 16 - 2 2 + 4 + 8 + 16 = 32 - 2 13. 111 , 3 = 37 222 , 6 = 37 333 , 9 = 37 14. Consider the following procedure:
Select a number. Double the number. Add 4 to the product.
Divide the sum by 2. Subtract 2 from the quotient.
a. Repeat the procedure for four numbers of your choice. Write a conjecture that relates the result of the process to
the original number selected.
b. Represent the original number by the variable n and use deductive reasoning to prove the conjecture in part (a).
1.2
15. The number 923,187,456 is called a pandigital square because it uses all the digits from 1 to 9 once each and is
the square of a number:
30,3842 = 30,384 * 30,384 = 923,187,456.
(Source: David Wells, The Penguin Dictionary of Curious and Interesting Numbers)
Round the pandigital square 923,187,456 to the nearest
a. hundred.
b. thousand.
c. hundred-thousand.
d. million.
e. hundred-million.
16. A magnified view of the boundary of this black “buglike” shape, called the Mandelbrot set, was illustrated in the
Section 1.1 opener on page 2.
3 units
2 units
The area of the blue rectangular region is the product of
its length, 3 units, and its width, 2 units, or 6 square units.
It is conjectured that the area of the black buglike region
representing the Mandelbrot set is
26p - 1 - e ≈ 1.5065916514855 square units.
(Source: Robert P. Munafo, Mandelbrot Set Glossary and Encyclopedia)
Round the area of the Mandelbrot set to
a. the nearest tenth.
b. the nearest hundredth.
c. the nearest thousandth.
d. seven decimal places.
In Exercises 17–20, obtain an estimate for each computation by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer?
17. 1.57 + 4.36 + 9.78 18. 8.83 * 49 19. 19.894 , 4.179 20. 62.3% of 3847.6
In Exercises 21–24, determine each estimate without using a calculator. Then use a calculator to perform the computation necessary to obtain an exact answer. How reasonable is your estimate when compared to the actual answer?
21. Estimate the total cost of six grocery items if their prices are $8.47, $0.89, $2.79, $0.14, $1.19, and $4.76.
22. Estimate the salary of a worker who works for 78 hours at $9.95 per hour.
23. At a yard sale, a person bought 21 books at $0.85 each, two chairs for $11.95 each, and a ceramic plate for $14.65.
Estimate the total amount spent.
24. The circle graph shows how the 20,207,375 students enrolled in U.S. colleges and universities in 2015 funded
college costs. Estimate the number of students who covered
these costs through grants and scholarships.
How Students Cover College Costs
30%
15% 12%
31%
Parental Loans
Relatives and Friends 4%
Student Loans
Student Income and Savings
Parental Income and Savings
Grants and Scholarships
8%
Source: The College Board
25. A small private school employs 10 teachers with salaries ranging from $817 to $992 per week. Which of the
following is the best estimate of the monthly payroll for the
teachers?
a. $30,000 b. $36,000
c. $42,000 d. $50,000
26. Select the best estimate for the number of seconds in a day.
a. 1500 b. 15,000
c. 86,000 d. 100,000
44 C H A P T E R 1 Problem Solving and Critical Thinking
Chapter Summary, Review, and Test 45
27. Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric
observations based on this scenario.
Earth’s Population as a Village of 200 People
150
50
75
100
125
25N u
m b
e r
o f
P e o
p le
Source: Gary Rimmer, Number Freaking, The Disinformation Company Ltd.
a. Which group in the village has a population that exceeds 100? Estimate this group’s population.
b. World population is approximately 33 million times the population of the village of 200 people. Use this
observation to estimate the number of people in the
world, in millions, unable to read or write.
28. The bar graph shows the percentage of people 25 years of age and older who were college graduates in the United
States for eight selected years.
1960
7.7
1970
11.0
1980
17.0
1990
21.3
2000
25.6
2010
29.9
36%
24%
28%
32%
16%
20%
12%
8%P e rc
e n
ta ge
W h
o W
e re
C o
ll e ge
G ra
d u
a te
s
Percentage of College Graduates, Among People Ages 25 and Older, in the United States
Year
1950
6.0
4%
2014
32.0
Source: U.S. Census Bureau
a. Estimate the average yearly increase in the percentage of college graduates. Round to the nearest tenth of a
percent.
b. If the trend shown by the graph continues, estimate the percentage of people 25 years of age and older who will
be college graduates in 2020.
29. During a diagnostic evaluation, a 33-year-old woman experienced a panic attack a few minutes after she had
been asked to relax her whole body. The graph at the top
of the next column shows the rapid increase in heart rate
during the panic attack.
120
100
80
60
110
90
70
H e
a rt
R a
te (
b e
a ts
p e
r m
in u
te )
Time (minutes)
Heart Rate before and during a Panic Attack
0 2 10 124 116 81 93 5 7
Source: Davis and Palladino, Psychology, Fifth Edition, Prentice Hall, 2007.
a. Use the graph to estimate the woman’s maximum heart rate during the first 12 minutes of the diagnostic
evaluation. After how many minutes did this occur?
b. Use the graph to estimate the woman’s minimum heart rate during the first 12 minutes of the diagnostic
evaluation. After how many minutes did this occur?
c. During which time period did the woman’s heart rate increase at the greatest rate?
d. After how many minutes was the woman’s heart rate approximately 75 beats per minute?
30. The bar graph shows the population of the United States, in millions, for five selected years.
P o
p u
la ti
o n
( m
il li
o n
s)
200
320
Population of the United States
Year
1980
226.5
1990
248.7
2000
281.4
2010
309.3
240
280
160
120
80
1970
203.3
40
Source: U.S. Census Bureau
a. Estimate the yearly increase in the U.S. population. Express the answer in millions and do not
round.
b. Write a mathematical model that estimates the U.S. population, p, in millions, x years after 1970.
c. Use the mathematical model from part (b) to project the U.S. population, in millions, in 2020.
46 C H A P T E R 1 Problem Solving and Critical Thinking
1.3
31. What necessary piece of information is missing that prevents solving the following problem?
If 3 milligrams of a medicine is given for every 20 pounds
of body weight, how many milligrams should be given to
a 6-year-old child?
32. In the following problem, there is one more piece of information given than is necessary for solving the problem.
Identify this unnecessary piece of information. Then solve
the problem.
A taxicab charges $3.00 for the first mile and $0.50 for
each additional half-mile. After a 6-mile trip, a customer
handed the taxi driver a $20 bill. Find the cost of the
trip.
Use the four-step method in problem solving to solve Exercises 33–39.
33. A company offers the following text message monthly price plans.
Pay-per-Text
$0.20 per regular text
$0.30 per photo or video text
Packages (include photo and video texts)
200 messages: $5.00 per month
1500 messages: $15.00 per month
Unlimited messages: $20.00 per month
Suppose that you send 40 regular texts and 35 photo texts
in a month. With which plan (pay-per-text or a package)
will you pay less money? How much will you save over the
other plan?
34. If there are seven frankfurters in one pound, how many pounds would you buy for a picnic to supply 28 people with
two frankfurters each?
35. A car rents for $175 per week plus $0.30 per mile. Find the rental cost for a three-week trip of 1200 miles.
36. The costs for two different kinds of heating systems for a two-bedroom home are given in the following table.
System Cost to install Operating cost
per year
Solar $29,700 $200
Electric $5500 $1800
After 12 years, which system will have the greater total
costs (installation cost plus operating cost)? How much
greater will the total costs be?
37. Miami is on Eastern Standard Time and San Francisco is on Pacific Standard Time, three hours earlier than Eastern
Standard Time. A flight leaves Miami at 10 a.m. Eastern
Standard Time, stops for 45 minutes in Houston, Texas,
and arrives in San Francisco at 1:30 p.m. Pacific time. What
is the actual flying time from Miami to San Francisco?
38. An automobile purchased for $37,000 is worth $2600 after eight years. Assuming that the value decreased steadily
each year, what was the car worth at the end of the fifth
year?
39. Suppose you are an engineer programming the automatic gate for a 35-cent toll. The gate is programmed for exact
change only and will not accept pennies. How many coin
combinations must you program the gate to accept?
1. Which reasoning process is shown in the following example?
The course policy states that if you turn in at least
80% of the homework, your lowest exam grade will
be dropped. I turned in 90% of the homework, so my
lowest grade will be dropped.
2. Which reasoning process is shown in the following example?
We examine the fingerprints of 1000 people. No two
individuals in this group of people have identical
fingerprints. We conclude that for all people, no two
people have identical fingerprints.
In Exercises 3–6, find the next number, computation, or figure, as appropriate.
3. 0, 5, 10, 15, ______ 4. 16 , 1 12 ,
1 24 ,
1 48 , _____
5. 3367 * 3 = 10,101 3367 * 6 = 20,202 3367 * 9 = 30,303 3367 * 12 = 40,404______________________
6. , , , , ,
Chapter 1 Test
7. Consider the following procedure:
Select a number. Multiply the number by 4. Add 8 to
the product. Divide the sum by 2. Subtract 4 from the
quotient.
a. Repeat this procedure for three numbers of your choice. Write a conjecture that relates the result of the
process to the original number selected.
b. Represent the original number by the variable n and use deductive reasoning to prove the conjecture in
part (a).
8. Round 3,279,425 to the nearest hundred-thousand.
9. Round 706.3849 to the nearest hundredth.
In Exercises 10–13, determine each estimate without using a calculator. Different rounding results in different estimates, so there is not one single correct answer to each exercise. Use rounding to make the resulting calculations simple.
10. For a spring break vacation, a student needs to spend $47.00 for gas, $311.00 for food, and $405.00 for a hotel
room. If the student takes $681.79 from savings, estimate
how much more money is needed for the vacation.
11. The cost for opening a restaurant is $485,000. If 19 people decide to share equally in the business, estimate the amount
each must contribute.
12. Find an estimate of 0.48992 * 121.976.
13. The graph shows the composition of a typical American community’s trash.
Paper 35%
Yard waste 12%
Food waste 12%
Plastic 11%
Metal 8%
Glass 5%
Other 17%
Types of Trash in an American Community by Percentage of Total Weight
Source: U.S. Environmental Protection Agency
Across the United States, people generate approximately
512 billion pounds of trash per year. Estimate the number
of pounds of trash in the form of plastic.
14. If the odometer of a car reads 71,911.5 miles and it averaged 28.9 miles per gallon, select the best estimate for
the number of gallons of gasoline used.
a. 2400 b. 3200 c. 4000 d. 4800 e. 5600
15. The stated intent of the 1994 “don’t ask, don’t tell” policy was to reduce the number of discharges of gay men and
lesbians from the military. Nearly 14,000 active-duty gay
servicemembers were dismissed under the policy, which
officially ended in 2011, after 18 years. The line graph at
the top of the next column shows the number of discharges
under “don’t ask, don’t tell” from 1994 through 2010.
N u
m b
e r
o f
D is
c h
a rg
e d
A c ti
v e -
D u
ty S
e rv
ic e m
e m
b e rs
Number of Active-Duty Gay Servicemembers Discharged from the Military for Homosexuality
Year
’10’94 ’95 ’96 ’97 ’98 ’99 ’00 ’01 ’02 ’03 ’04 ’05 ’06 ’07 ’08 ’09
200
300
400
500
600
700
800
900
1000
1100
1200
1300
100
Source: General Accountability Office
a. For the period shown, in which year did the number of discharges reach a maximum? Find a reasonable
estimate of the number of discharges for that
year.
b. For the period shown, in which year did the number of discharges reach a minimum? Find a reasonable
estimate of the number of discharges for that year.
c. In which one-year period did the number of discharges decrease at the greatest rate?
d. In which year were approximately 1000 gay service- members discharged under the “don’t ask, don’t tell”
policy?
16. Grade Inflation. The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school.
60%
50%
40%
30%
20%
Percentage of U.S. College Freshmen with an Average Grade of A (A− to A+) in High School
Year
2013
53%
2010
48%
2000
43%
1990
29%
1980
27%
10%
P e rc
e n
ta g e o
f C
o ll
e g e F
re sh
m e n
w it
h a
n A
H ig
h S
c h
o o
l A
v e ra
g e
Source: Higher Education Research Institute
a. Estimate the average yearly increase in the percentage of high school grades of A. Round to the nearest tenth
of a percent.
b. Write a mathematical model that estimates the percentage of high school grades of A, p, x years after 1980.
c. If the trend shown by the graph continues, use your mathematical model from part (b) to project the
percentage of high school grades of A in 2020.
Chapter 1 Test 47
48 C H A P T E R 1 Problem Solving and Critical Thinking
17. The cost of renting a boat from Estes Rental is $9 per 15 minutes. The cost from Ship and Shore Rental is $20
per half-hour. If you plan to rent the boat for three hours,
which business offers the better deal and by how much?
18. A bus operates between Miami International Airport and Miami Beach, 10 miles away. It makes 20 round trips per
day carrying 32 passengers per trip. If the fare each way
is $11.00, how much money is taken in from one day’s
operation?
19. By paying $50 cash up front and the balance at $35 a week, how long will it take to pay for a computer costing
$960?
20. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per
year. In the same year, the population of Belgium was
10,200,000, with projections of a population decrease of
12,000 people per year. (Source: United Nations) According to these projections, which country will
have the greater population in 2035 and by how many more
people?
Set Theory OUR BODIES ARE FRAGILE AND COMPLEX, VULNERABLE TO DISEASE AND EASILY
DAMAGED. THE SEQUENCING OF THE HUMAN GENOME IN 2003—ALL 140,000 GENES—
should lead to rapid advances in treating heart disease, cancer, depression,
Alzheimer’s, and AIDS. Neural stem cell research could make it possible to repair
brain damage and even re-create whole parts of the brain. There appears to be
no limit to the parts of our bodies that can be replaced. By contrast, at the start
of the twentieth century, we lacked even a basic understanding of the different
types of human blood. The discovery of blood types, organized into collections
called sets and illustrated by a special set diagram, rescued surgery patients from
random, often lethal, transfusions. In this sense, the set diagram for blood types
that you will encounter in this chapter reinforces our optimism that life does
improve and that we are better off today than we were one hundred years ago.
2
Here’s where you’ll find this application: Organizing and visually representing sets of human blood
types is presented in the Blitzer Bonus on page 94. The vital
role that this representation plays in blood transfusions is
developed in Exercises 113–117 of Exercise Set 2.4.
49
50 C H A P T E R 2 Set Theory
2.1 Basic Set Concepts WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Use three methods to represent sets.
2 Define and recognize the empty set.
3 Use the symbols ∊ and ∉. 4 Apply set notation to sets of
natural numbers.
5 Determine a set’s cardinal number.
6 Recognize equivalent sets. 7 Distinguish between finite and
infinite sets.
8 Recognize equal sets.
1 Use three methods to represent sets.
EXAMPLE 1 Representing a Set Using a Description
Write a word description of the set
P = 5Washington, Adams, Jefferson, Madison, Monroe6.
SOLUTION
Set P is the set of the first five presidents of the United States.
CHECK POINT 1 Write a word description of the set L = 5a, b, c, d, e, f6.
WE TEND TO PLACE THINGS IN
categories, which allows us
to order and structure the
world. For example, to which
populations do you belong?
Do you categorize yourself
as a college student? What
about your gender? What
about your academic major
or your ethnic background?
Our minds cannot find order
and meaning without creating
collections. Mathematicians call
such collections sets. A set is a collection of objects whose contents can be clearly determined. The objects in a set
are called the elements, or members, of the set. A set must be well defined, meaning that its contents can be clearly determined.
Using this criterion, the collection of actors who have won Academy Awards is a
set. We can always determine whether or not a particular actor is an element of this
collection. By contrast, consider the collection of great actors. Whether or not a
person belongs to this collection is a matter of how we interpret the word great. In this text, we will only consider collections that form well-defined sets.
Methods for Representing Sets
An example of a set is the set of the days of the week, whose elements are Monday,
Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.
Capital letters are generally used to name sets. Let’s use W to represent the set of the days of the week.
Three methods are commonly used to designate a set. One method is a word description. We can describe set W as the set of the days of the week. A second method is the roster method. This involves listing the elements of a set inside a pair of braces, 5 6. The braces at the beginning and end indicate that we are representing a set. The roster form uses commas to separate the elements of the
set. Thus, we can designate the set W by listing its elements:
W = 5Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday6.
Grouping symbols such as parentheses, 1 2, and square brackets, 3 4, are not used to represent sets. Only commas are used to separate the elements of a set.
Separators such as colons or semicolons are not used. Finally, the order in which
the elements are listed in a set is not important. Thus, another way of expressing
the set of the days of the week is
W = 5Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday6.
S E C T I O N 2 . 1 Basic Set Concepts 51
EXAMPLE 2 Representing a Set Using the Roster Method
Set C is the set of U.S. coins with a value of less than a dollar. Express this set using the roster method.
SOLUTION
C = 5penny, nickel, dime, quarter, half@dollar6
CHECK POINT 2 Set M is the set of months beginning with the letter A. Express this set using the roster method.
The third method for representing a set is with set-builder notation. Using this method, the set of the days of the week can be expressed as
W x
W = 5 x 0 x is a day of the week6.
We read this notation as “Set W is the set of all elements x such that x is a day of the week.” Before the vertical line is the variable x, which represents an element in general. After the vertical line is the condition x must meet in order to be an element of the set.
Table 2.1 contains two examples of sets, each represented with a word description, the roster method, and set-builder notation.
Do I have to use x to represent the variable in set-builder notation?
No. Any letter can be used to
represent the variable. Thus,
5x� x is a day of the week6, 5y� y is a day of the week6, and 5z� z is a day of the week6 all represent the same set.
GREAT QUESTION!
T A B L E 2 . 1 Sets Using Three Designations
Word Description Roster Method Set-Builder Notation
B is the set of members of the Beatles in 1963.
B = 5George Harrison, John Lennon, Paul
McCartney, Ringo Starr6
B = 5x� x was a member of the Beatles in 19636
S is the set of states whose names begin with the letter A.
S = 5Alabama, Alaska, Arizona, Arkansas6
S = 5x� x is a U.S. state whose name begins with
the letter A6
The Beatles climbed to the top of the
British music charts in 1963, conquering
the United States a year later.
EXAMPLE 3 Converting from Set-Builder to Roster Notation
Express set
A = 5x� x is a month that begins with the letter M6
using the roster method.
SOLUTION
Set A is the set of all elements x such that x is a month beginning with the letter M. There are two such months, namely March and May. Thus,
A = 5March, May6.
CHECK POINT 3 Express the set O = 5x� x is a positive odd number less than 106
using the roster method.
52 C H A P T E R 2 Set Theory
The representation of some sets by the roster method can be rather long, or
even impossible, if we attempt to list every element. For example, consider the set
of all lowercase letters of the English alphabet. If L is chosen as a name for this set, we can use set-builder notation to represent L as follows:
L = 5x� x is a lowercase letter of the English alphabet6.
A complete listing using the roster method is rather tedious:
L = 5a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z6.
We can shorten the listing in set L by writing
L = 5a, b, c, d, c, z6.
The three dots after the element d, called an ellipsis, indicate that the elements in the set continue in the same manner up to and including the last element z.
Have you ever considered what would happen if we suddenly lost our ability to recall
categories and the names that identify them? This is precisely what happened to
Alice, the heroine of Lewis Carroll’s Through the Looking Glass, as she walked with a fawn in “the woods with no names.”
So they walked on together through the woods, Alice with her arms clasped
lovingly round the soft neck of the Fawn, till they came out into another open
field, and here the Fawn gave a sudden bound into the air, and shook itself free
from Alice’s arm. “I’m a Fawn!” it cried out in a voice of delight. “And, dear me!
you’re a human child!” A sudden look of alarm came into its beautiful brown
eyes, and in another moment it had darted away at full speed.
By realizing that Alice is a member of the set of human beings, which in turn
is part of the set of dangerous things, the fawn is overcome by fear. Thus, the fawn’s
experience is determined by the way it structures the world into sets with various
characteristics.
Blitzer Bonus The Loss of Sets
The Empty Set
Consider the following sets:
5x� x is a fawn that speaks6 5x� x is a number greater than 10 and less than 46.
Can you see what these sets have in common? They both contain no elements.
There are no fawns that speak. There are no numbers that are both greater than 10
and also less than 4. Sets such as these that contain no elements are called the empty set, or the null set.
2 Define and recognize the empty set.
T H E E M P T Y S E T
The empty set, also called the null set, is the set that contains no elements. The empty set is represented by 5 6 or ∅.
Notice that 5 6 and ∅ have the same meaning. However, the empty set is not represented by 5∅6. This notation represents a set containing the element ∅.
S E C T I O N 2 . 1 Basic Set Concepts 53
EXAMPLE 4 Recognizing the Empty Set
Which one of the following is the empty set?
a. 506 b. 0
c. 5x� x is a number less than 4 or greater than 106 d. 5x� x is a square with exactly three sides6
SOLUTION
a. 506 is a set containing one element, 0. Because this set contains an element, it is not the empty set.
b. 0 is a number, not a set, so it cannot possibly be the empty set. It does, however, represent the number of members of the empty set.
c. 5x� x is a number less than 4 or greater than 106 contains all numbers that are either less than 4, such as 3, or greater than 10, such as 11.
Because some elements belong to this set, it cannot be the empty set.
d. 5x� x is a square with exactly three sides6 contains no elements. There are no squares with exactly three sides. This set is the empty set.
CHECK POINT 4 Which one of the following is the empty set? a. 5x� x is a number less than 3 or greater than 56 b. 5x� x is a number less than 3 and greater than 56 c. nothing
d. 5∅6
Notations for Set Membership
We now consider two special notations that indicate whether or not a given object
belongs to a set.
John Cage (1912–1992), the
American avant-garde composer,
translated the empty set into
the quietest piece of music ever
written. His piano composition
4′33″ requires the musician to sit frozen in silence at a piano
stool for 4 minutes, 33 seconds,
or 273 seconds. (The significance
of 273 is that at approximately
- 273°C, all molecular motion stops.) The set
5x� x is a musical sound from 4′33″6
is the empty set. There are
no musical sounds in the
composition. Mathematician
Martin Gardner wrote, “I have
not heard 4′33″ performed, but friends who have tell me it is
Cage’s finest composition.”
Blitzer Bonus The Musical Sounds of the Empty Set
3 Use the symbols ∊ and ∉.
T H E N O TAT I O N S ∊ A N D ∉ The symbol ∊ is used to indicate that an object is an element of a set. The symbol ∊ is used to replace the words “is an element of.” The symbol ∉ is used to indicate that an object is not an element of a set. The symbol ∉ is used to replace the words “is not an element of.”
EXAMPLE 5 Using the Symbols ∊ and ∉
Determine whether each statement is true or false:
a. r∊5a, b, c, c, z6 b. 7∉51, 2, 3, 4, 56 c. 5a6∊5a, b6.
SOLUTION
a. Because r is an element of the set 5a, b, c, c, z6, the statement
r∊5a, b, c, c, z6
is true.
Observe that an element can belong to a set in roster notation when
three dots appear even though the element is not listed.
54 C H A P T E R 2 Set Theory
CHECK POINT 5 Determine whether each statement is true or false: a. 8∊51, 2, 3, c, 106 b. r∉5a, b, c, z6 c. 5Monday6∊5x� x is a day of the week6.
b. Because 7 is not an element of the set 51, 2, 3, 4, 56, the statement
7∉51, 2, 3, 4, 56 is true.
c. Because 5a6 is a set and the set 5a6 is not an element of the set 5a, b6, the statement
5a6∊5a, b6 is false.
Sets of Natural Numbers
For much of the remainder of this section, we will focus on the set of numbers used
for counting:
51, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, c6.
The set of counting numbers is also called the set of natural numbers. We represent this set by the bold face letter N.
Can a set ever belong to another set—sort of a set within a set?
Yes. A set can be an element
of another set. For example,
55a, b6, c6 is a set with two elements. One element is
the set 5a, b6 and the other element is the letter c. Thus, 5a, b6∊55a, b6, c6 and c∊55a, b6, c6.
GREAT QUESTION!
4 Apply set notation to sets of natural numbers.
T H E S E T O F N AT U R A L N U M B E R S
N = 51, 2, 3, 4, 5, c6
The three dots, or ellipsis, after the 5 indicate that there is no final element and that
the listing goes on forever.
EXAMPLE 6 Representing Sets of Natural Numbers
Express each of the following sets using the roster method:
a. Set A is the set of natural numbers less than 5.
b. Set B is the set of natural numbers greater than or equal to 25.
c. E = 5x� x∊N and x is even6.
SOLUTION
a. The natural numbers less than 5 are 1, 2, 3, and 4. Thus, set A can be expressed using the roster method as
A = 51, 2, 3, 46.
b. The natural numbers greater than or equal to 25 are 25, 26, 27, 28, and so on. Set B in roster form is
B = 525, 26, 27, 28, c6.
The three dots show that the listing goes on forever.
c. The set-builder notation
E = 5x� x∊N and x is even6
indicates that we want to list the set of all x such that x is an element of the set of natural numbers and x is even. The set of numbers that meets both conditions is the set of even natural numbers. The set in roster form is
E = 52, 4, 6, 8, c6.
S E C T I O N 2 . 1 Basic Set Concepts 55
CHECK POINT 6 Express each of the following sets using the roster method: a. Set A is the set of natural numbers less than or equal to 3.
b. Set B is the set of natural numbers greater than 14.
c. O = 5x� x∊N and x is odd6.
A BRIEF REVIEW Inequality Notation Inequality symbols are frequently used to describe sets of natural numbers. Table 2.2 reviews basic inequality notation.
x a
x a
x a
x a
x a b
x a
b
x a
b
x a
b
x
x 6 a
a 6 x 6 b
a … x … b
a 6 x … b
a … x 6 b
x 7 a
x Ú a
x … a
51, 2, 36
55, 6, 76
51, 2, 3, 46
55, 6, 7, 8, …6
55, 6, 7, 86
54, 5, 6, 7, …6
54, 5, 6, 76
54, 5, 6, 7, 86
5x 0x H N and x 6 46
5x 0x H N and x … 46
5x 0x H N and x 7 46
5x 0x H N and x Ú 46
5x 0x H N and 4 6 x 6 86
5x 0x H N and 4 … x … 86
5x 0x H N and 4 … x 6 86
5x 0x H N and 4 6 x … 86
x
x
x
x
x
x
x
Roster MethodSet-Builder Notation Inequality Symbol
and Meaning Example
T A B L E 2 . 2 Inequality Notation and Sets
56 C H A P T E R 2 Set Theory
EXAMPLE 7 Representing Sets of Natural Numbers
Express each of the following sets using the roster method:
a. 5x� x∊N and x … 1006 b. 5x� x∊N and 70 … x 6 1006.
SOLUTION
a. 5x� x∊N and x … 1006 represents the set of natural numbers less than or equal to 100. This set can be expressed using the roster method as
51, 2, 3, 4, c, 1006.
b. 5x� x∊N and 70 … x 6 1006 represents the set of natural numbers greater than or equal to 70 and less than 100. This set in roster form is
570, 71, 72, 73, c, 996.
CHECK POINT 7 Express each of the following sets using the roster method: a. 5x� x∊N and x 6 2006 b. 5x� x∊N and 50 6 x … 2006.
5 Determine a set’s cardinal number. Cardinality and Equivalent Sets The number of elements in a set is called the cardinal number, or cardinality, of the set. For example, the set 5a, e, i, o, u6 contains five elements and therefore has the cardinal number 5. We can also say that the set has a cardinality of 5.
D E F I N I T I O N O F A S E T ’ S C A R D I N A L N U M B E R
The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. The symbol n(A) is read “n of A.”
Notice that the cardinal number of a set refers to the number of distinct, or different, elements in the set. Repeating elements in a set neither adds new elements to the set nor changes its cardinality. For example, A = 53, 5, 76 and B = 53, 5, 5, 7, 7, 76 represent the same set with three distinct elements, 3, 5, and 7. Thus, n(A) = 3 and n(B) = 3.
EXAMPLE 8 Determining a Set’s Cardinal Number
Find the cardinal number of each of the following sets:
a. A = 57, 9, 11, 136 b. B = 506 c. C = 513, 14, 15, c, 22, 236 d. ∅.
SOLUTION
The cardinal number for each set is found by determining the number of
elements in the set.
a. A = 57, 9, 11, 136 contains four distinct elements. Thus, the cardinal number of set A is 4. We also say that set A has a cardinality of 4, or n(A) = 4.
b. B = 506 contains one element, namely, 0. The cardinal number of set B is 1. Therefore, n(B) = 1.
S E C T I O N 2 . 1 Basic Set Concepts 57
c. Set C = 513, 14, 15, c, 22, 236 lists only five elements. However, the three dots indicate that the natural numbers from 16 through 21 are also
in the set. Counting the elements in the set, we find that there are
11 natural numbers in set C. The cardinality of set C is 11, and n(C) = 11. d. The empty set, ∅, contains no elements. Thus, n(∅) = 0.
CHECK POINT 8 Find the cardinal number of each of the following sets: a. A = 56, 10, 14, 15, 166 b. B = 58726 c. C = 59, 10, 11, c, 15, 166 d. D = 5 6.
Sets that contain the same number of elements are said to be equivalent.6 Recognize equivalent sets. D E F I N I T I O N O F E Q U I VA L E N T S E T S
Set A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets, n(A) = n(B).
Here is an example of two equivalent sets:
n A = n B = A = 5x 0x is a vowel6 = 5a, e, i, o, u6
B = 5x 0x H N and 3 … x … 76 = 53, 4, 5, 6, 76.
It is not necessary to count elements and arrive at 5 to determine that these sets are
equivalent. The lines with arrowheads, D , indicate that each element of set A can be paired with exactly one element of set B and each element of set B can be paired with exactly one element of set A. We say that the sets can be placed in a one-to-one correspondence.
O N E - T O - O N E C O R R E S P O N D E N C E S A N D E Q U I VA L E N T S E T S
1. If set A and set B can be placed in a one-to-one correspondence, then A is equivalent to B: n(A) = n(B).
2. If set A and set B cannot be placed in a one-to-one correspondence, then A is not equivalent to B: n(A) ≠ n(B).
EXAMPLE 9 Determining If Sets Are Equivalent
Figure 2.1 shows the top five impediments to academic performance for U.S. college students.
30%
25%
20%
15%
10%
P e rc
e n
ta ge
o f
S tu
d e n
ts R
e p
o rt
in g
E a c h
I m
p e d
im e n
t
Top Five Impediments to Academic Performance
Impediment to Academic Performance
Stress Sleep Problems
Illness Anxiety Work
14
1919 20
28
5%
F I G U R E 2 . 1
Source: American College Health Association
58 C H A P T E R 2 Set Theory
Let
A = the set of five impediments shown in Figure 2.1 B = the set of the percentage of college students
reporting each impediment.
Are these sets equivalent? Explain.
SOLUTION
Let’s begin by expressing each set in roster form.
A = 5stress, sleep problems, illness, anxiety, work6
B = 5 28, 20, 19, 14 6
There are two ways to determine that these sets are not equivalent.
Method 1. Trying to Set Up a One-to-One Correspondence
The lines with arrowheads between the sets in roster form indicate that
the correspondence between the sets is not one-to-one. The elements
illness and anxiety from set A are both paired with the element 19 from set B. These sets are not equivalent.
Method 2. Counting Elements
Set A contains five distinct elements: n(A) = 5. Set B contains four distinct elements: n(B) = 4. Because the sets do not contain the same number of elements, they are not equivalent.
CHECK POINT 9 Figure 2.2 shows the percentage of Americans optimistic about the future for each region of the country. Let
A = the set of the four regions shown in Figure 2.2 B = the set of the percentage of Americans in each
region optimistic about the future.
Are these sets equivalent? Explain.
80%
70%
60%
50%
40%
30%
P er
ce n
ta ge
O p
ti m
is ti
c A
b o
u t
th e
F u
tu re
Percentage of Americans Optimistic About the Future
Region Northeast
68
South
73
Midwest
73
West
75
20%
F I G U R E 2 . 2
Source: The Harris Poll (2016 data)
30%
25%
20%
15%
10%
P er
ce n
ta ge
o f
S tu
d en
ts R
ep o
rt in
g E
ac h
I m
p ed
im en
t
Top Five Impediments to Academic Performance
Impediment to Academic Performance
St re
ss Sl
ee p
Pr ob
le m
s Ill
ne ss
A nx
ie ty
W or
k
14
191920
28
5%
F I G U R E 2 . 1 (repeated)
S E C T I O N 2 . 1 Basic Set Concepts 59
An example of an infinite set is the set of natural numbers,
N = 51, 2, 3, 4, 5, 6, c6, where the ellipsis indicates that there is no last, or final, element. Does this set have a cardinality? The answer is yes, albeit one of
the strangest numbers you’ve ever seen. The set of natural numbers is assigned
the infinite cardinal number ℵ0 (read: “aleph-null,” aleph being the first letter of the Hebrew alphabet). What follows is a succession of mind-boggling results,
including a hierarchy of different infinite numbers in which ℵ0 is the smallest infinity:
ℵ0 6 ℵ1 6 ℵ2 6 ℵ3 6 ℵ4 6 ℵ5 c.
These ideas, which are impossible for our imaginations to grasp, are developed in
Section 2.2 and the Blitzer Bonus at the end of that section.
7 Distinguish between finite and infinite sets. Finite and Infinite Sets Example 9 illustrated that to compare the cardinalities of two sets, pair off their
elements. If there is not a one-to-one correspondence, the sets have different
cardinalities and are not equivalent. Although this idea is obvious in the case of
finite sets, some unusual conclusions emerge when dealing with infinite sets.
F I N I T E S E T S A N D I N F I N I T E S E T S
Set A is a finite set if n(A) = 0 (that is, A is the empty set) or n(A) is a natural number. A set whose cardinality is not 0 or a natural number is called an infinite set.
8 Recognize equal sets. Equal Sets We conclude this section with another important concept of set theory, equality of
sets.
D E F I N I T I O N O F E Q U A L I T Y O F S E T S
Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize
the equality of sets A and B using the statement A = B.
For example, if A = 5w, x, y, z6 and B = 5z, y, w, x6, then A = B because the two sets contain exactly the same elements.
Because equal sets contain the same elements, they also have the same cardinal
number. For example, the equal sets A = 5w, x, y, z6 and B = 5z, y, w, x6 have four elements each. Thus, both sets have the same cardinal number: 4. Notice
that a possible one-to-one correspondence between the equal sets A and B can be obtained by pairing each element with itself:
A = 5w, x, y, z6
B = 5z, y, w, x6
This illustrates an important point: If two sets are equal, then they must be equivalent.
Can you clarify the difference between equal sets and equivalent sets?
In English, the words equal and equivalent often mean the same thing. This is not the
case in set theory. Equal sets contain the same elements. Equivalent sets contain the same number of elements. If two sets are equal, then they
must be equivalent. However,
if two sets are equivalent, they
are not necessarily equal.
GREAT QUESTION!
60 C H A P T E R 2 Set Theory
CHECK POINT 10 Determine whether each statement is true or false: a. 5O, L, D6 = 5D, O, L6 b. 54, 56 = 55, 4, ∅6.
Fill in each blank so that the resulting statement is true.
1. The set 5California, Colorado, Connecticut6 is expressed using the ________ method. The set 5x|x is a U.S. state whose name begins with the letter C6 is expressed using ____________ notation.
2. A set that contains no elements is called the null set or the ________ set. This set is represented by 5 6 or _____.
3. The symbol ∊ is used to indicate that an object ______________ of a set.
4. The set N = 51, 2, 3, 4, 5, c6 is called the set of _________________.
5. The number of distinct elements in a set is called the __________ number of the set. If A represents the set, this number is represented by ________.
6. Two sets that contain the same number of elements are called ____________ sets.
7. Two sets that contain the same elements are called ________ sets.
Concept and Vocabulary Check
Practice Exercises
In Exercises 1–6, determine which collections are not well defined and therefore not sets.
Exercise Set 2.1
EXAMPLE 10 Determining Whether Sets Are Equal
Determine whether each statement is true or false:
a. 54, 8, 96 = 58, 9, 46
b. 51, 3, 56 = 50, 1, 3, 56.
SOLUTION
a. The sets 54, 8, 96 and 58, 9, 46 contain exactly the same elements. Therefore, the statement
54, 8, 96 = 58, 9, 46 is true.
b. As we look at the given sets, 51, 3, 56 and 50, 1, 3, 56, we see that 0 is an element of the second set, but not the first. The sets do not contain
exactly the same elements. Therefore, the sets are not equal. This means
that the statement
51, 3, 56 = 50, 1, 3, 56 is false.
1. The collection of U.S. presidents
2. The collection of part-time and full-time students currently enrolled at your college
3. The collection of the five worst U.S. presidents
4. The collection of elderly full-time students currently enrolled at your college
5. The collection of natural numbers greater than one million
6. The collection of even natural numbers greater than 100
S E C T I O N 2 . 1 Basic Set Concepts 61
In Exercises 7–14, write a word description of each set. (More than one correct description may be possible.)
7. 5Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune6
8. 5Saturday, Sunday6
9. 5January, June, July6
10. 5April, August6
11. 56, 7, 8, 9, c6
12. 59, 10, 11, 12, c6
13. 56, 7, 8, 9, c, 206
14. 59, 10, 11, 12, c, 256
In Exercises 15–32, express each set using the roster method.
15. The set of the four seasons in a year
16. The set of months of the year that have exactly 30 days
17. 5x� x is a month that ends with the letters b@e@r6 18. 5x� x is a lowercase letter of the alphabet that follows d and
comes before j6
19. The set of natural numbers less than 4
20. The set of natural numbers less than or equal to 6
21. The set of odd natural numbers less than 13
22. The set of even natural numbers less than 10
23. 5x� x∊N and x … 56 24. 5x� x∊N and x … 46 25. 5x� x∊N and x 7 56 26. 5x� x∊N and x 7 46 27. 5x� x∊N and 6 6 x … 106 28. 5x� x∊N and 7 6 x … 116 29. 5x� x∊N and 10 … x 6 806 30. 5x� x∊N and 15 … x 6 606 31. 5x� x + 5 = 76 32. 5x� x + 3 = 96
In Exercises 33–46, determine which sets are the empty set.
33. 5∅, 06 34. 50, ∅6 35. 5x� x is a woman who served as U.S. president before
20166
36. 5x� x is a living U.S. president born before 12006 37. 5x� x is the number of women who served as U.S. president
before 20166
38. 5x� x is the number of living U.S. presidents born before 12006
39. 5x� x is a U.S. state whose name begins with the letter X6 40. 5x� x is a month of the year whose name begins with the
letter X6
41. 5x� x 6 2 and x 7 56
42. 5x� x 6 3 and x 7 76 43. 5x� x∊N and 2 6 x 6 56 44. 5x� x∊N and 3 6 x 6 76 45. 5x� x is a number less than 2 or greater than 56 46. 5x� x is a number less than 3 or greater than 76
In Exercises 47–66, determine whether each statement is true or false.
47. 3∊51, 3, 5, 76 48. 6∊52, 4, 6, 8, 106 49. 12∊51, 2, 3, c, 146 50. 10∊51, 2, 3, c, 166 51. 5∊52, 4, 6, c, 206 52. 8∊51, 3, 5, c196 53. 11∉51, 2, 3, c, 96 54. 17∉51, 2, 3, c, 166 55. 37∉51, 2, 3, c, 406 56. 26∉51, 2, 3, c, 506 57. 4∉5x� x∊N and x is even6 58. 2∊5x� x∊N and x is odd6 59. 13∉5x� x∊N and x 6 136 60. 20∉5x� x∊N and x 6 206 61. 16∉5x� x∊N and 15 … x 6 206 62. 19∉5x� x∊N and 16 … x 6 216 63. 536∊53, 46 64. 576∊57, 86 65. - 1 ∉ N 66. - 2 ∉ N
In Exercises 67–80, find the cardinal number for each set.
67. A = 517, 19, 21, 23, 256 68. A = 516, 18, 20, 22, 24, 266 69. B = 52, 4, 6, c, 306 70. B = 51, 3, 5, c, 216 71. C = 5x� x is a day of the week that begins with the letter A6 72. C = 5x� x is a month of the year that begins with the
letter W6
73. D = 5five6 74. D = 5six6 75. A = 5x� x is a letter in the word five6 76. A = 5x� x is a letter in the word six6 77. B = 5x� x∊N and 2 … x 6 76 78. B = 5x� x∊N and 3 … x 6 106 79. C = 5x� x 6 4 and x Ú 126 80. C = 5x� x 6 5 and x Ú 156
62 C H A P T E R 2 Set Theory
In Exercises 81–90,
a. Are the sets equivalent? Explain.
b. Are the sets equal? Explain.
81. A is the set of students at your college. B is the set of students majoring in business at your college.
82. A is the set of states in the United States. B is the set of people who are now governors of the states in the United
States.
83. A = 51, 2, 3, 4, 56 B = 50, 1, 2, 3, 46 84. A = 51, 3, 5, 7, 96 B = 52, 4, 6, 8, 106 85. A = 51, 1, 1, 2, 2, 3, 46 B = 54, 3, 2, 16 86. A = 50, 1, 1, 2, 2, 2, 3, 3, 3, 36 B = 53, 2, 1, 06 87. A = 5x� x∊N and 6 … x 6 106 B = 5x� x∊N and 9 6 x … 136 88. A = 5x� x∊N and 12 6 x … 176 B = 5x� x∊N and 20 … x 6 256 89. A = 5x� x∊N and 100 … x … 1056 B = 5x� x∊N and 99 6 x 6 1066 90. A = 5x� x∊N and 200 … x … 2066 B = 5x� x∊N and 199 6 x 6 2076 In Exercises 91–96, determine whether each set is finite or infinite.
91. 5x� x∊N and x Ú 1006 92. 5x� x∊N and x Ú 506 93. 5x� x∊N and x … 1,000,0006 94. 5x� x∊N and x … 2,000,0006 95. The set of natural numbers less than 1
96. The set of natural numbers less than 0
Practice Plus
In Exercises 97–100, express each set using set-builder notation. Use inequality notation to express the condition x must meet in order to be a member of the set. (More than one correct inequality may be possible.)
97. 561, 62, 63, 64, c6
98. 536, 37, 38, 39, c6
99. 561, 62, 63, 64, c, 896
100. 536, 37, 38, 39, c, 596
In Exercises 101–104, give examples of two sets that meet the given conditions. If the conditions are impossible to satisfy, explain why.
101. The two sets are equivalent but not equal.
102. The two sets are equivalent and equal.
103. The two sets are equal but not equivalent.
104. The two sets are neither equivalent nor equal.
Application Exercises
Although you want to choose a career that fits your interests and abilities, it is good to have an idea of what jobs pay when looking at career options. The bar graph shows the average yearly earnings of full-time employed college graduates with only a bachelor’s degree based on their college major.
$80
A v e ra
g e Y
e a rl
y E
a rn
in g s
(t h
o u
sa n
d s
o f
d o
ll a rs
)
Average Earnings, by College Major
38
S o
c ia
l W
o rk
43
P h
il o
so p
h y
51
N u
rs in
g
53 57
63
76
J o
u rn
a li
sm
M a rk
e ti
n g
A c c o
u n
ti n
g
$10
$20
$30
$40
$50
$60
$70
E n
g in
e e ri
n g
Source: Arthur J. Keown, Personal Finance, Pearson
In Exercises 105–108, use the information given by the graph to represent each set by the roster method.
105. The set of college majors with average yearly earnings that exceed $57,000
106. The set of college majors with average yearly earnings that exceed $63,000
107. {x� x is a major with $38,000 6 average yearly earnings … $53,000}
108. {x� x is a major with $38,000 … average yearly earnings 6 $53,000}
The bar graph shows the differences among age groups on the Implicit Association Test that measures levels of racial prejudice. Higher scores indicate stronger bias.
46
44
40
36
32
S c o
re o
n t
h e I
m p
li c it
A ss
o c ia
ti o
n T
e st
Measuring Racial Prejudice, by Age
Age Range
42
34
31
29
32
35
33
28
42
38
34
30
26
: little or no bias
: slight bias
: moderate bias
Key: <15 15–35
36–65
Below
18
18–24 25–34 35–44 45–54 55–64 65+
Source: The Race Implicit Association Test on the Project Implicit Demonstration Website
S E C T I O N 2 . 1 Basic Set Concepts 63
In Exercises 109–112, use the information given by the graph at the bottom of the previous page to represent each set by the roster method, or use the appropriate notation to indicate that the set is the empty set.
109. {x� x is a group whose score indicates little or no bias}
110. {x� x is a group whose score indicates slight bias}
111. {x� x is a group whose score indicates moderate bias}
112. {x� x is a group whose score is at least 30 and at most 40}
A study of 900 working women in Texas showed that their feelings changed throughout the day. The following line graph shows 15 different times in a day and the average level of happiness for the women at each time. Based on the information given by the graph, represent each of the sets in Exercises 113–116 using the roster method.
5
4
3
2
222120191817161514131211109
A v e ra g e L
e v e l
o f
H a p
p in
e ss
Time of Day
Average Level of Happiness at Different Times of Day
8
1
Source: D. Kahneman et al. “A Survey Method for Characterizing Daily Life Experience,” Science
113. 5x� x is a time of the day when the average level of happiness was 36
114. 5x� x is a time of the day when the average level of happiness was 16
115. 5x� x is a time of the day when
3 6 average level of happiness 6 46 116. 5x� x is a time of the day when
3 6 average level of happiness … 46 117. Do the results of Exercise 113 or 114 indicate a one-to-one
correspondence between the set representing the time of
day and the set representing average level of happiness?
Are these sets equivalent?
Explaining the Concepts
118. What is a set?
119. Describe the three methods used to represent a set. Give an example of a set represented by each method.
120. What is the empty set?
121. Explain what is meant by equivalent sets.
122. Explain what is meant by equal sets.
123. Use cardinality to describe the difference between a finite set and an infinite set.
Critical Thinking Exercises
Make Sense? In Exercises 124–127, determine whether each statement makes sense or does not make sense, and explain your reasoning.
124. I used the roster method to express the set of countries that I have visited.
125. I used the roster method and natural numbers to express the set of average daily Fahrenheit temperatures throughout the
month of July in Vostok Station, Antarctica, the coldest month
in one of the coldest locations in the world.
126. Using this bar graph that shows the average number of hours that Americans sleep per day, I can see that there is
a one-to-one correspondence between the set of six ages
on the horizontal axis and the set of the average number of
hours that men sleep per day.
H o
u rs
S le
p t
p e r
D a y
Hours Slept per Day, by Age
Age
17
8.0
9.7
9.3
22
8.7
9.1
30
8.4
8.8
40
8.3 8.5
50
8.2 8.4
60
8.3 8.5
Men Women
8.4
8.8
9.2
9.6
10.0
Source: ATUS, Bureau of Labor Statistics
127. Using the bar graph in Exercise 126, I can see that there is a one-to-one correspondence between the set of the average
number of hours that men sleep per day and the set of the
average number of hours that women sleep per day.
In Exercises 128–135, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
128. Two sets can be equal but not equivalent.
129. Any set in roster notation that contains three dots must be an infinite set.
130. n(∅) = 1 131. Some sets that can be written in set-builder notation cannot
be written in roster form.
132. The set of fractions between 0 and 1 is an infinite set.
133. The set of multiples of 4 between 0 and 4,000,000,000 is an infinite set.
134. If the elements in a set cannot be counted in a trillion years, the set is an infinite set.
135. Because 0 is not a natural number, it can be deleted from any set without changing the set’s cardinality.
136. In a certain town, a barber shaves all those men and only those men who do not shave themselves. Consider each of
the following sets:
A = 5x� x is a man of the town who shaves himself6 B = 5x� x is a man of the town who does not shave himself6. The one and only barber in the town is Sweeney Todd. If s
represents Sweeney Todd,
a. is s∊A? b. is s∊B?
64 C H A P T E R 2 Set Theory
T A B L E 2 . 3 Percentage of Tattooed
Americans, by Age Group
Age Group Percent Tattooed
18–24 22%
25–29 30%
30–39 38%
40–49 27%
50–64 11%
65 + 5%
Source: Harris Interactive
Subsets2.2 WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Recognize subsets and use the notation ⊆.
2 Recognize proper subsets and use the notation ⊂.
3 Determine the number of subsets of a set.
4 Apply concepts of subsets and equivalent sets to infinite sets.
MATH TATTOOS. WHO KNEW? EMERGING
from their often unsavory reputation of the
recent past, tattoos have gained increasing
prominence as a form of body art and self-
expression. A recent Harris poll estimated
that 45 million Americans, or 21% of the
adult population, have at least one tattoo.
Table 2.3 shows the percentage of Americans, by age group, with tattoos.
The categories in the table divide the set
of tattooed Americans into smaller sets,
called subsets, based on age. The age subsets can be broken into still-smaller
subsets. For example, tattooed Americans
ages 25–29 can be categorized by gender,
political party affiliation, race/ethnicity, or
any other area of interest. This suggests numerous possible subsets of the set of
Americans with tattoos. Every American in each of these subsets is also a member
of the set of tattooed Americans.
Subsets
Situations in which all the elements of one set are also elements of another set are
described by the following definition:
1 Recognize subsets and use the notation ⊆.
D E F I N I T I O N O F A S U B S E T O F A S E T
Set A is a subset of set B, expressed as
A ⊆ B,
if every element in set A is also an element in set B.
Let’s apply this definition to the set of people ages 25–29 in Table 2.3.
5x 0x is a tattooed American and 25 … x’s age … 296
5x 0x is a tattooed American68
Observe that a subset is itself a set.
The notation A h B means that A is not a subset of B. Set A is not a subset of set B if there is at least one element of set A that is not an element of set B. For example, consider the following sets:
A = 51, 2, 36 and B = 51, 26.
Can you see that 3 is an element of set A that is not in set B? Thus, set A is not a subset of set B: A h B.
We can show that A ⊆ B by showing that every element of set A also occurs as an element of set B. We can show that A h B by finding one element of set A that is not in set B.
S E C T I O N 2 . 2 Subsets 65
Earth
Venus Mercury
Mars
Jupiter
Saturn
Uranus
Neptune
The eight planets in Earth’s solar system
No, we did not forget Pluto. In 2006,
based on the requirement that a planet
must dominate its own orbit (Pluto is
slave to Neptune’s orbit), the International
Astronomical Union removed Pluto from
the list of planets and decreed that it
belongs to a new category of heavenly
body, a “dwarf planet.”
EXAMPLE 1 Using the Symbols ⊆ and h
Write ⊆ or h in each blank to form a true statement: a. A = 51, 3, 5, 76
B = 51, 3, 5, 7, 9, 116 A B
b. A = 5x� x is a letter in the word proof6 B = 5y� y is a letter in the word roof6 A B
c. A = 5x� x is a planet of Earth>s solar system6 B = 5Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune6 A B
SOLUTION
a. All the elements of A = 51, 3, 5, 76 are also contained in B = 51, 3, 5, 7, 9, 116. Therefore, set A is a subset of set B:
A ⊆ B.
b. Let’s write the set of letters in the word proof and the set of letters in the word roof in roster form. In each case, we consider only the distinct elements, so there is no need to repeat the o.
A = 5p, r, o, f6 B = 5r, o, f6
p A B
Because there is an element in set A that is not in set B, set A is not a subset of set B:
A h B. c. All the elements of
A = 5x� x is a planet of Earth>s solar system6
are contained in
B = 5Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune6.
Because all elements in set A are also in set B, set A is a subset of set B:
A ⊆ B.
Furthermore, the sets are equal (A = B).
CHECK POINT 1 Write ⊆ or h in each blank to form a true statement: a. A = 51, 3, 5, 6, 9, 116
B = 51, 3, 5, 76 A B
b. A = 5x� x is a letter in the word roof6 B = 5y� y is a letter in the word proof6 A B
c. A = 5x� x is a day of the week6 B = 5Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday6 A B
66 C H A P T E R 2 Set Theory
Try not to confuse the symbols for subset, ⊆, and proper subset, ⊂. In some subset examples, both symbols can be placed between sets:
51, 36 8 51, 3, 56 and
A B
A B A B
51, 36 ( 51, 3, 56.
A B
A B A B
By contrast, there are subset examples where only the symbol ⊆ can be placed between sets:
51, 3, 56 8 51, 3, 56.
A B
A B A B A
B A = B (
Because the lower part of the subset symbol in A ⊆ B suggests an equal sign, it is possible that sets A and B are equal, although they do not have to be. By contrast, the missing lower line for the proper subset symbol in A ⊂ B indicates that sets A and B cannot be equal.
Proper Subsets
In Example 1(c) and Check Point 1(c), the given sets are equal and illustrate that
every set is a subset of itself. If A is any set, then A ⊆ A because it is obvious that each element of A is a member of A.
If we know that set A is a subset of set B and we exclude the possibility of the sets being equal, then set A is called a proper subset of set B, written A ⊂ B.
2 Recognize proper subsets and use the notation ⊂.
D E F I N I T I O N O F A P R O P E R S U B S E T O F A S E T
Set A is a proper subset of set B, expressed as A ⊂ B, if set A is a subset of set B and sets A and B are not equal (A ≠ B).
Is there a relationship between the symbols # and ⊂ and the inequality symbols " and *?
Great observation!
• The notation for “is a subset
of,” ⊆, is similar to the notation for “is less than or
equal to,” … . Because the notations share similar ideas,
A ⊆ B applies to finite sets only if the cardinal number
of set A is less than or equal to the cardinal number of
set B. • The notation for “is a proper
subset of,” ⊂, is similar to the notation for “is less than,” 6 . Because the notations share
similar ideas, A ⊂ B applies to finite sets only if the
cardinal number of set A is less than the cardinal
number of set B.
GREAT QUESTION!
EXAMPLE 2 Using the Symbols ⊆ and ⊂
Write ⊆, ⊂, or both, in each blank to form a true statement:
a. A = 5x� x is a person and x lives in San Francisco6 B = 5x� x is a person and x lives in California6 A B
b. A = 52, 4, 6, 86 B = 52, 8, 4, 66 A B.
SOLUTION
a. We begin with A = 5x 0x is a person and x lives in San Francisco6 and B = 5x � x is a person and x lives in California6. Every person living in San Francisco is also a person living in California. Because each person
in set A is contained in set B, set A is a subset of set B:
A ⊆ B.
S E C T I O N 2 . 2 Subsets 67
Can you see that the two sets, A = 5x 0x is a person and x lives in San Francisco6 and B = 5x � x is a person and x lives in California6, do not contain exactly the same elements and, consequently, are not equal?
A person living in California outside San Francisco is in set B, but not in set A. Because there is at least one such person, the sets are not equal and set A is a proper subset of set B:
A ⊂ B.
The symbols ⊆ and ⊂ can both be placed in the blank to form a true statement.
b. Every number in A = 52, 4, 6, 86 is contained in B = 52, 8, 4, 66, so set A is a subset of set B:
A ⊆ B.
Because the sets contain exactly the same elements and are equal, set A is not a proper subset of set B. The symbol ⊂ cannot be placed in the blank if we want to form a true statement. (Because set A is not a proper subset of set B, it is correct to write A ⊄ B.)
CHECK POINT 2 Write ⊆, ⊂, or both in each blank to form a true statement: a. A = 52, 4, 6, 86
B = 52, 8, 4, 6, 106 A B
b. A = 5x� x is a person and x lives in Atlanta6 B = 5x� x is a person and x lives in Georgia6 A B
All the symbols used in set theory make me feel that I’m an element of the set of the notationally confused! For example, what’s the difference between the symbols { and #?
The symbol ∊ means “is an element of” and the symbol ⊆ means “is a subset of.” Notice the differences among the following true statements:
H
546 x 54, 86.4 H 54, 86 546 8 54, 86
GREAT QUESTION!
We opened the section by considering subsets of the set of
tattooed Americans, based on age. We’ll continue dividing the
set of tattooed Americans into subsets using party affiliation
and gender in the Exercise Set at the end of this section (see
Exercises 83–92).
In Science Ink (Sterling, 2011), science writer Carl Zimmer presents more than 300 thought-provoking science
and math tattoos, explaining the significance of the body art.
Many of the tattooed images in Zimmer’s book relate to topics
you’ll encounter in Thinking Mathematically, including the empty set, numerals in base two (Section 4.2), the golden ratio
(Section 6.5), and Σ, a symbol of summation, that appears
in many statistical formulas
(Section 12.2). Check out
Science Ink and prepare to be dazzled by the
images and the
stories behind
them.
Blitzer Bonus Science and Math Tattoos
68 C H A P T E R 2 Set Theory
Example 3 illustrates the principle that the empty set is a subset of every set. Furthermore, the empty set is a proper subset of every set except itself.
Subsets and the Empty Set
The meaning of A ⊆ B leads to some interesting properties of the empty set.
EXAMPLE 3 The Empty Set as a Subset
Let A = 5 6 and B = 51, 2, 3, 4, 56. Is A ⊆ B?
SOLUTION
A is not a subset of B (A h B) if there is at least one element of set A that is not an element of set B. Because A represents the empty set, there are no elements in set A, period, much less elements in A that do not belong to B. Because we cannot find an element in A = 5 6 that is not contained in B = 51, 2, 3, 4, 56, this means that A ⊆ B. Equivalently, ∅ ⊆ B.
CHECK POINT 3 Let A = 5 6 and B = 56, 7, 86. Is A ⊆ B?
T H E E M P T Y S E T A S A S U B S E T
1. For any set B, ∅ ⊆ B. 2. For any set B other than the empty set, ∅ ⊂ B.
The Number of Subsets of a Given Set
If a set contains n elements, how many distinct subsets can be formed? Let’s observe some special cases, namely sets with 0, 1, 2, and 3 elements. We can use
inductive reasoning to arrive at a general conclusion. We begin by listing subsets
and counting the number of subsets in our list. This is shown in Table 2.4.
3 Determine the number of subsets of a set.
T A B L E 2 . 4 The Number of Subsets: Some Special Cases
Set Number of Elements List of All Distinct Subsets
Number of Subsets
5 6 0 5 6 1
5a6 1 5a6, 5 6 2
5a, b6 2 5a, b6, 5a6, 5b6, 5 6 4
5a, b, c6 3 5a, b, c6,
5a, b6, 5a, c6, 5b, c6,
5a6, 5b6, 5c6, 5 6
8
Table 2.4 suggests that when we increase the number of elements in the set by one, the number of subsets doubles. The number of subsets appears to be a
power of 2.
Number of elements 0 1 2 3
Number of subsets 1 = 20 2 = 21 4 = 2 * 2 = 22 8 = 2 * 2 * 2 = 23
S E C T I O N 2 . 2 Subsets 69
The Number of Subsets of Infinite Sets
In Section 2.1, we mentioned that the infinite set of natural numbers,
51, 2, 3, 4, 5, 6, c6, is assigned the cardinal number ℵ0 (read “aleph-null”), called a transfinite cardinal number. Equivalently, there are ℵ0 natural numbers.
Once we accept the cardinality of sets with infinitely many elements, a surreal
world emerges in which there is no end to an ascending hierarchy of infinities.
Because the set of natural numbers contains ℵ0 elements, it has 2 ℵ
0 subsets, where
2ℵ0 7 ℵ0 . Denoting 2ℵ0 by ℵ1 , we have ℵ1 7 ℵ0 . Because the set of subsets of the natural numbers contains ℵ1 elements, it has 2
ℵ 1 subsets, where 2ℵ1 7 ℵ1 . Denoting
2ℵ1 by ℵ2 , we now have ℵ2 7 ℵ1 7 ℵ0 . Continuing in this manner, ℵ0 is the “smallest” transfinite cardinal number in an infinite hierarchy of different infinities!
The power of 2 is the same as the number of elements in the set. Using inductive
reasoning, if the set contains n elements, then the number of subsets that can be formed is 2n.
A BRIEF REVIEW Powers of 2 If powers of 2 have you in an
exponentially increasing state
of confusion, here’s a list of
values that should be helpful.
Observe how rapidly these
values are increasing.
Powers of 2 20 = 1 21 = 2 22 = 2 * 2 = 4 23 = 2 * 2 * 2 = 8 24 = 2 * 2 * 2 * 2 = 16 25 = 2 * 2 * 2 * 2 * 2 = 32 26 = 64 27 = 128 28 = 256 29 = 512
210 = 1024 211 = 2048 212 = 4096 215 = 32,768 220 = 1,048,576 225 = 33,554,432 230 = 1,073,741,824
N U M B E R O F S U B S E T S
The number of distinct subsets of a set with n elements is 2n.
N U M B E R O F P R O P E R S U B S E T S
The number of distinct proper subsets of a set with n elements is 2n - 1.
For a given set, we know that every subset except the set itself is a proper
subset. In Table 2.4, we included the set itself when counting the number of subsets. If we want to find the number of proper subsets, we must exclude counting the
given set, thereby decreasing the number by 1.
EXAMPLE 4 Finding the Number of Subsets and Proper Subsets
Find the number of distinct subsets and the number of distinct proper subsets
for each set:
a. 5a, b, c, d, e6
b. 5x� x∊N and 9 … x … 156.
SOLUTION
a. A set with n elements has 2n subsets. Because the set 5a, b, c, d, e6 contains 5 elements, there are 25 = 2 * 2 * 2 * 2 * 2 = 32 subsets. Of these, we must exclude counting the given set as a proper subset, so
there are 25 - 1 = 32 - 1 = 31 proper subsets. b. We can write 5x� x∊N and 9 … x … 156 in roster form as 59, 10, 11, 12, 13, 14, 156. Because this set contains 7 elements, there are 27 = 2 * 2 * 2 * 2 * 2 * 2 * 2 = 128 subsets. Of these, there are 27 - 1 = 128 - 1 = 127 proper subsets.
CHECK POINT 4 Find the number of distinct subsets and the number of distinct proper subsets for each set:
a. 5a, b, c, d6
b. 5x� x∊N and 3 … x … 86.
4 Apply concepts of subsets and equivalent sets to infinite sets.
“Infinity is where things happen that don’t.” —W. W. Sawyer, Prelude to Mathematics, Penguin Books, 1960
70 C H A P T E R 2 Set Theory
The mirrors in the painting Time and Time Again have the effect of repeating the image infinitely many times, creating
an endless tunnel of mirror images. There is something quite
fascinating about the idea of endless infinity. Did you know
that for thousands of years religious leaders warned that
human beings should not examine the nature of the infinite?
Religious teaching often equated infinity with the concept of
a Supreme Being. One of the last victims of the Inquisition,
Giordano Bruno, was burned at the stake for his explorations
into the characteristics of infinity. It was not until the 1870s
that the German mathematician Georg Cantor (1845–1918)
began a careful analysis of the mathematics of infinity.
It was Cantor who assigned the transfinite cardinal
number ℵ0 to the set of natural numbers. He used one-to-one correspondences to establish some surprising equivalences
between the set of natural numbers and its proper subsets.
Here are two examples:
Blitzer Bonus Cardinal Numbers of Infinite Sets
Time and Time Again (1981), P.J. Crook/Bridgeman Art Library
Natural Numbers: 51, 2, 3, 4, 5 , 6 , …, n, …6
Even Natural Numbers: 52, 4, 6, 8, 10, 12, …, 2n, …6
Natural Numbers: 51, 2, 3, 4, 5, 6, …, n, …6
Odd Natural Numbers: 51, 3, 5, 7, 9, 11, …, 2n - 1, …6
n n n n -
These one-to-one correspondences indicate that the set of even natural numbers and the set of odd natural numbers are
equivalent to the set of all natural numbers. In fact, an infinite set, such as the natural numbers, can be defined as any set that can be placed in a one-to-one correspondence with a proper subset of itself. This definition boggles the mind because it implies
that part of a set has the same number of objects as the entire set. There are ℵ0 even natural numbers, ℵ0 odd natural numbers, and ℵ0 natural numbers. Because the even and odd natural numbers combined make up the entire set of natural numbers, we are confronted with an unusual statement of transfinite arithmetic:
ℵ0 + ℵ0 = ℵ0 .
As Cantor continued studying infinite sets, his observations grew stranger and stranger. It was Cantor who showed that
some infinite sets contain more elements than others. This was too much for his colleagues, who considered this work ridiculous.
Cantor’s mentor, Leopold Kronecker, told him, “Look at the crazy ideas that are now surfacing with your work with infinite sets.
How can one infinity be greater than another? Best to ignore such inconsistencies. By considering these monsters and infinite
numbers mathematics, I will make sure that you never gain a faculty position at the University of Berlin.” Although Cantor was
not burned at the stake, universal condemnation of his work resulted in numerous nervous breakdowns. His final days, sadly, were
spent in a psychiatric hospital. However, Cantor’s work later regained the respect of mathematicians. Today, he is seen as a great
mathematician who demystified infinity.
Fill in each blank so that the resulting statement is true.
1. Set A is a subset of set B, expressed as _________, means that _______________________________________________.
2. Set A is a proper subset of set B, expressed as _________, means that set A is a subset of set B and __________________________.
3. The statement ∅ ⊆ B tells us that ____________ set is a ________ of every set.
4. The number of distinct subsets of a set with n elements is _____.
5. The number of distinct proper subsets of a set with n elements is _________.
Concept and Vocabulary Check
S E C T I O N 2 . 2 Subsets 71
Practice Exercises
In Exercises 1–18, write ⊆ or h in each blank so that the resulting statement is true.
1. 51, 2, 56 _____ 51, 2, 3, 4, 5, 6, 76
2. 52, 3, 76 _____ 51, 2, 3, 4, 5, 6, 76
3. 5- 3, 0, 36 _____ 5- 3, - 1, 1, 36 4. 5- 4, 0, 46 _____ 5- 4, - 3, - 1, 1, 3, 46 5. 5Monday, Friday6 _____ 5Saturday, Sunday, Monday, Tuesday, Wednesday6
6. 5Mercury, Venus, Earth6 _____ 5Venus, Earth, Mars, Jupiter6
7. 5x� x is a cat6 _____ 5x� x is a black cat6 8. 5x� x is a dog6 _____ 5x� x is a pure@bred dog6 9. 5c, o, n, v, e, r, s, a, t, i, o, n6 _____ 5v, o, i, c, e, s, r, a, n, t, o, n6
10. 5r, e, v, o, l, u, t, i, o, n6 _____ 5t, o, l, o, v, e, r, u, i, n6
11. 547 , 9 136 _____ 5
7 4 ,
13 9 6 12. 5
1 2 ,
1 36 _____ 52, 3, 56
13. ∅ _____ 52, 4, 66 14. ∅ _____ 51, 3, 56 15. 52, 4, 66 _____ ∅ 16. 51, 3, 56 _____ ∅ 17. 5 6 _____ ∅ 18. ∅ _____ 5 6
In Exercises 19–40, determine whether ⊆, ⊂, both, or neither can be placed in each blank to form a true statement.
19. 5V, C, R6 _____ 5V, C, R, S6
20. 5F, I, N6 _____ 5F, I, N, K6
21. 50, 2, 4, 6, 86 _____ 58, 0, 6, 2, 46
22. 59, 1, 7, 3, 46 _____ 51, 3, 4, 7, 96
23. 5x� x is a man6 _____ 5x� x is a woman6 24. 5x� x is a woman6 _____ 5x� x is a man6 25. 5x� x is a man6 _____ 5x� x is a person6 26. 5x� x is a woman6 _____ 5x� x is a person6 27. 5x� x is a man or a woman6 _____ 5x� x is a person6 28. 5x� x is a woman or a man6 _____ 5x� x is a person6 29. A = 5x� x∊N and 5 6 x 6 126
B = the set of natural numbers between 5 and 12 A _____ B
30. A = 5x� x∊N and 3 6 x 6 106 B = the set of natural numbers between 3 and 10 A _____ B
31. A = 5x� x∊N and 5 6 x 6 126 B = the set of natural numbers between 3 and 17 A _____ B
32. A = 5x� x∊N and 3 6 x 6 106 B = the set of natural numbers between 2 and 16 A _____ B
33. A = 5x� x∊N and 5 6 x 6 126 B = 5x� x∊N and 2 … x … 116 A _____ B
34. A = 5x� x∊N and 3 6 x 6 106 B = 5x� x∊N and 2 … x … 86 A _____ B
35. ∅ _____ 57, 8, 9, c , 1006 36. ∅ _____ 5101, 102, 103, c , 2006 37. 57, 8, 9, c6 _____ ∅ 38. 5101, 102, 103, c6 _____ ∅ 39. ∅ _____ 5 6 40. 5 6 _____ ∅
In Exercises 41–54, determine whether each statement is true or false. If the statement is false, explain why.
41. Ralph∊5Ralph, Alice, Trixie, Norton6 42. Canada∊5Mexico, United States, Canada6 43. Ralph ⊆ 5Ralph, Alice, Trixie, Norton6 44. Canada ⊆ 5Mexico, United States, Canada6 45. 5Ralph6 ⊆ 5Ralph, Alice, Trixie, Norton6 46. 5Canada6 ⊆ 5Mexico, United States, Canada6 47. ∅∊5Archie, Edith, Mike, Gloria6 48. ∅ ⊆ 5Charlie Chaplin, Groucho Marx, Woody Allen6 49. 556∊5556, 5966 50. 516∊5516, 5366 51. 51, 46 h 54, 16 52. 51, 46 ⊄ 54, 16 53. 0∉∅ 54. 506 h ∅
In Exercises 55–60, list all the subsets of the given set.
55. 5border collie, poodle6 56. 5Romeo, Juliet6
57. 5t, a, b6 58. 5I, II, III6
59. 506 60. ∅
In Exercises 61–68, calculate the number of distinct subsets and the number of distinct proper subsets for each set.
61. 52, 4, 6, 86 62. 512 , 1 3 ,
1 4 ,
1 56
63. 52, 4, 6, 8, 10, 126 64. 5a, b, c, d, e, f6
65. 5x� x is a day of the week6 66. 5x� x is a U.S. coin worth less than a dollar6 67. 5x� x∊N and 2 6 x 6 66 68. 5x� x∊N and 2 … x … 66
Practice Plus
In Exercises 69–82, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
69. The set 51, 2, 3, c, 10006 has 21000 proper subsets.
70. The set 51, 2, 3, c, 10,0006 has 210,000 proper subsets.
71. 5x� x∊N and 30 6 x 6 506⊆5x 0x∊N and 30 … x … 506 72. 5x� x∊N and 20 … x … 606 h 5x 0x∊N and 20 6 x 6 606 73. ∅ h 5∅, 5∅66 74. 5∅6 h 5∅, 5∅66 75. ∅∊5∅, 5∅66
Exercise Set 2.2
72 C H A P T E R 2 Set Theory
76. 5∅6∊5∅, 5∅66 77. If A ⊆ B and d∊A, then d∊B. 78. If A ⊆ B and B ⊆ C, then A ⊆ C. 79. If set A is equivalent to the set of natural numbers, then
n(A) = ℵ0 .
80. If set A is equivalent to the set of even natural numbers, then n(A) = ℵ0 .
81. The set of subsets of 5a, e, i, o, u6 contains 64 elements.
82. The set of subsets of 5a, b, c, d, e, f6 contains 128 elements.
Application Exercises
We opened this section citing a Harris poll that estimated 45 million Americans have at least one tattoo. The bar graph on the left shows the percentage of tattooed Americans, by party affiliation and gender.
Number of Tattooed Americans per 10,000 Adults, by Party Affiliation and Gender
1890 Tattooed Americans
per 10,000 Adults
699 Democrats
529 Republicans
662 Independents
315 Men
384 Women
238 Men
291 Women
298 Men
364 Women
Source: Harris Interactive
Sets and subsets allow us to order and structure the data. On the right, the set of tattooed Americans is divided into subsets categorized by party affiliation. These subsets are further broken down into subsets categorized by gender. All numbers in the branching tree diagram are based on the number of people per 10,000 American adults. Based on the tree diagram, let
T = the set of tattooed Americans R = the set of tattooed Republicans D = the set of tattooed Democrats M = the set of tattooed Democratic men W = the set of tattooed Democratic women.
In Exercises 83–92, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
83. D∊T 84. R∊T 85. M ⊂ T 86. W ⊂ T 87. If x∊D, then x∊W. 88. If x∊D, then x∊M. 89. If x∊R, then x ∉ D. 90. If x∊D, then x ∉ R. 91. The set of elements in M and W combined is equal to
set D.
92. The set of elements in M and W combined is equivalent to set D.
93. Houses in Euclid Estates are all identical. However, a person can purchase a new house with some, all, or none
of a set of options. This set includes 5pool, screened-in balcony, lake view, alarm system, upgraded landscaping6. How many options are there for purchasing a house in this
community?
94. A cheese pizza can be ordered with some, all, or none of the following set of toppings: 5beef, ham, mushrooms, sausage, peppers, pepperoni, olives, prosciutto, onion6. How many different variations are available for ordering a pizza?
95. Based on more than 1500 ballots sent to film notables, the American Film Institute rated the top U.S. movies. The
Institute selected Citizen Kane (1941), The Godfather (1972), Casablanca (1942), Raging Bull (1980), Singin’ in the Rain (1952), and Gone with the Wind (1939) as the top six films. Suppose that you have all six films on DVD and decide to
view some, all, or none of these films. How many viewing
options do you have?
96. A small town has four police cars. If a radio dispatcher receives a call, depending on the nature of the situation, no
cars, one car, two cars, three cars, or all four cars can be sent.
How many options does the dispatcher have for sending the
police cars to the scene of the caller?
97. According to the U.S. Census Bureau, the most ethnically diverse U.S. cities are New York City, Los Angeles, Miami,
Chicago, Washington, D.C., Houston, San Diego, and Seattle.
If you decide to visit some, all, or none of these cities, how
many travel options do you have?
98. Film documentaries with the highest box office grosses include
Fahrenheit 9/11 ($222 million), March of the Penguins ($127 million), Earth ($109 million), Justin Bieber: Never Say Never ($99 million), Oceans ($83 million), One Direction: This Is Us ($69 million), and Bowling for Columbine ($58 million). (Source: Top 10 of Everything 2017, Portable Press)
Suppose that you have all seven documentaries on DVD
and decide, over the course of a week, to view some, all,
or none of these films. How many viewing options do you
have?
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 73
Explaining the Concepts
99. Explain what is meant by a subset.
100. What is the difference between a subset and a proper subset?
101. Explain why the empty set is a subset of every set.
102. Describe the difference between the symbols ∊ and ⊆. Explain how each symbol is used.
103. Describe the formula for finding the number of distinct subsets for a given set. Give an example.
104. Describe how to find the number of distinct proper subsets for a given set. Give an example.
Critical Thinking Exercises
Make Sense? In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.
105. The set of my six rent payments from January through June is a subset of the set of my 12 cable television payments
from January through December.
106. Every time I increase the number of elements in a set by one, I double the number of distinct subsets.
107. Because Exercises 93–98 involve different situations, I cannot solve them by the same method.
108. I recently purchased a set of books and am deciding which books, if any, to take on vacation. The number of subsets
of my set of books gives me the number of different
combinations of the books that I can take.
In Exercises 109–112, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
109. The set 536 has 23, or eight, subsets. 110. All sets have subsets. 111. Every set has a proper subset. 112. The set 53, 51, 466 has eight subsets. 113. Suppose that a nickel, a dime, and a quarter are on a table.
You may select some, all, or none of the coins. Specify all of
the different amounts of money that can be selected.
114. If a set has 127 proper subsets, how many elements are there in the set?
Group Exercises
115. This activity is a group research project and should result in a presentation made by group members to the entire class.
Georg Cantor was certainly not the only genius in history
who faced criticism during his lifetime, only to have his
work acclaimed as a masterpiece after his death. Describe
the life and work of three other people, including at least
one mathematician, who faced similar circumstances.
116. Research useful websites and present a report on infinite sets and their cardinalities. Explain why the sets of whole
numbers, integers, and rational numbers each have cardinal
number ℵ0 . Be sure to define these sets and show the one-to- one correspondences between each set and the set of natural
numbers. Then explain why the set of real numbers does not
have cardinal number ℵ0 by describing how a real number can always be left out in a pairing with the natural numbers. Spice
up the more technical aspects of your report with ideas you
discovered about infinity that you find particularly intriguing.
Venn Diagrams and Set Operations2.3 WHAT AM I
SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Understand the meaning of a universal set.
2 Understand the basic ideas of a Venn diagram.
3 Use Venn diagrams to visualize relationships between two sets.
4 Find the complement of a set. 5 Find the intersection of two
sets.
6 Find the union of two sets. 7 Perform operations with sets. 8 Determine sets involving
set operations from a Venn
diagram.
9 Understand the meaning of and and or.
10 Use the formula for n(A ∪ B).
LATINOS MAKE UP APPROXIMATELY 17% OF THE U.S.
population and pump an estimated $1.3 trillion into the
economy each year, equal to the GDP of Mexico, the
Dominican Republic, Guatemala, and El Salvador
combined. (Source: Time) As Latino spending power steadily rises, corporate America has discovered that
Hispanic Americans, particularly young spenders between
the ages of 14 and 34, want to be spoken to in English,
even as they stay true to their Latino identity.
What is the primary language spoken at home by
U.S. Hispanics? In this section, we use sets to analyze
the answer to this question. By doing so, you will
see how sets and their visual representations
provide precise ways of organizing, classifying, and
describing a wide variety of data.
Universal Sets and Venn Diagrams
The circle graph in Figure 2.3 on the next page categorizes America’s 55 million Hispanics by the
primary language spoken at home. The graph’s
sectors define four sets:
• the set of U.S. Hispanics who speak Spanish at home.
• the set of U.S. Hispanics who speak English at home.
74 C H A P T E R 2 Set Theory
• the set of U.S. Hispanics who speak both Spanish and English at home.
• the set of U.S. Hispanics who speak neither
Spanish nor English at home.
In discussing sets, it is convenient to refer
to a general set that contains all elements under
discussion. This general set is called the universal set. A universal set, symbolized by U, is a set that contains all the elements being considered in a
given discussion or problem. Thus, a convenient
universal set for the sets described above is
U = the set of U.S. Hispanics.
Notice how this universal set restricts our attention
so that we can divide it into the four subsets shown
by the circle graph in Figure 2.3. We can obtain a more thorough understanding
of sets and their relationship to a universal set by considering diagrams that
allow visual analysis. Venn diagrams, named for the British logician John Venn (1834–1923), are used to show the visual relationship among sets.
Figure 2.4 is a Venn diagram. The universal set is represented by a region inside a rectangle. Subsets within
the universal set are depicted by circles, or sometimes
by ovals or other shapes. In this Venn diagram, set A is represented by the light blue region inside the circle.
The dark blue region in Figure 2.4 represents the set of elements in the universal set U that are not in set A. By combining the regions shown by the light blue shading
and the dark blue shading, we obtain the universal set, U.
1 Understand the meaning of a universal set.
English 32%
Both 23%
Spanish 43%
Other 2%
Languages Spoken at Home by U.S. Hispanics
F I G U R E 2 . 3
Source: Time
2 Understand the basic ideas of a Venn diagram.
A U
F I G U R E 2 . 4
Is the size of the circle in a Venn diagram important?
No. The size of the circle
representing set A in a Venn diagram has nothing to do with
the number of elements in
set A.
GREAT QUESTION!
A
$ M 5
U
F I G U R E 2 . 5
A 1 5 6
7 9
U
F I G U R E 2 . 6
EXAMPLE 1 Determining Sets from a Venn Diagram
Use the Venn diagram in Figure 2.5 to determine each of the following sets:
a. U b. A c. the set of elements in U that are not in A.
SOLUTION
a. Set U, the universal set, consists of all the elements within the rectangle. Thus, U = 5□, △, $, M, 56.
b. Set A consists of all the elements within the circle. Thus, A = 5□, △6. c. The set of elements in U that are not in A, shown by the set of all the
elements outside the circle, is 5$, M, 56.
CHECK POINT 1 Use the Venn diagram in Figure 2.6 to determine each of the following sets:
a. U b. A
c. the set of elements in U that are not in A.
3 Use Venn diagrams to visualize relationships between two sets. Representing Two Sets in a Venn Diagram There are a number of different ways to represent two subsets of a universal set in
a Venn diagram. To help understand these representations, consider the following
scenario:
You need to determine whether there is sufficient support on campus to have a
blood drive. You take a survey to obtain information, asking students
Would you be willing to donate blood?
Would you be willing to help serve a free breakfast to blood donors?
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 75
Set A represents the set of students willing to donate blood. Set B represents the set of students willing to help serve breakfast to donors. Possible survey results include
the following:
• No students willing to donate blood are willing to serve breakfast, and vice
versa.
• All students willing to donate blood are willing to serve breakfast.
• The same students who are willing to donate blood are willing to serve
breakfast.
• Some of the students willing to donate blood are willing to serve breakfast.
We begin by using Venn diagrams to visualize these results. To do so, we consider
four basic relationships and their visualizations.
Relationship 1: Disjoint Sets Two sets that have no elements in common are called disjoint sets. Two disjoint sets, A and B, are shown in the Venn diagram in Figure 2.7. Disjoint sets are represented as circles that do not overlap. No elements of set A are elements of set B, and vice versa.
Since set A represents the set of students willing to donate blood and set B represents the set of students willing to serve breakfast to donors, the set
diagram illustrates
No students willing to donate blood are willing to serve breakfast, and
vice versa.
Relationship 2: Proper Subsets If set A is a proper subset of set B (A ⊂ B), the relationship is shown in the Venn diagram in Figure 2.8. All elements of set A are elements of set B. If an x representing an element is placed inside circle A, it automatically falls inside circle B.
Since set A represents the set of students willing to donate blood and set B represents the set of students willing to serve breakfast to donors, the set
diagram illustrates
All students willing to donate blood are willing to serve breakfast.
Relationship 3: Equal Sets If A = B, then set A contains exactly the same elements as set B. This relationship is shown in the Venn diagram in Figure 2.9. Because all elements in set A are in set B, and vice versa, this diagram illustrates that when A = B, then A ⊆ B and B ⊆ A.
Since set A represents the set of students willing to donate blood and set B represents the set of students willing to serve breakfast to donors, the set
diagram illustrates
The same students who are willing to donate blood are willing to serve
breakfast.
Relationship 4: Sets with Some Common Elements In mathematics, the word some means there exists at least one. If set A and set B have at least one element in common, then the circles representing the sets must overlap. This is illustrated
in the Venn diagram in Figure 2.10. Since set A represents the set of students willing to donate blood and
set B represents the set of students willing to serve breakfast to donors, the presence of at least one student in the dark blue region in Figure 2.10 illustrates
Some students willing to donate blood are willing to serve breakfast.
In Figure 2.11 at the top of the next page, we’ve numbered each of the regions in the Venn diagram in Figure 2.10. Let’s make sure we understand what these regions represent in terms of the campus blood drive scenario. Remember that A is the set of blood donors and B is the set of breakfast servers.
A B U
F I G U R E 2 . 7
B
A
U
F I G U R E 2 . 8
A = B U
F I G U R E 2 . 9
U A B
Common elements are in this region.
F I G U R E 2 . 1 0
76 C H A P T E R 2 Set Theory
Region II This region represents the set of students willing to donate blood and serve
breakfast. The elements that belong to both set A and set B are in this region.
Region I This region represents the set of students willing to donate blood but not
serve breakfast. The elements that belong to set A but not to set B are in this region.
Region III This region represents the set of students willing to serve breakfast but
not donate blood. The elements that belong to set B but not to set A are in this region.
Region IV This region represents the set of students surveyed who are not willing
to donate blood and are not willing to serve breakfast. The elements that
belong to the universal set U that are not in sets A or B are in this region.
In Figure 2.11, we’ll start with the innermost region, region II, and work outward to region IV.
U
A B
A: Set of blood donors B: Set of breakfast servers
IV
I II III
F I G U R E 2 . 1 1 EXAMPLE 2 Determining Sets from a Venn Diagram
Use the Venn diagram in Figure 2.12 to determine each of the following sets:
a. U b. B
c. the set of elements in A but not B
d. the set of elements in U that are not in B
e. the set of elements in both A and B.
SOLUTION
a. Set U, the universal set, consists of all elements within the rectangle. Taking the elements in regions I, II, III, and IV, we obtain U = 5a, b, c, d, e, f, g6.
b. Set B consists of the elements in regions II and III. Thus, B = 5d, e6. c. The set of elements in A but not B, found in region I, is 5a, b, c6.
d. The set of elements in U that are not in B, found in regions I and IV, is 5a, b, c, f, g6.
e. The set of elements in both A and B, found in region II, is 5d6.
II
A U
B
III
IV f g
I a b c
d e
F I G U R E 2 . 1 2
CHECK POINT 2 Use the Venn diagram in Figure 2.12 to determine each of the following sets:
a. A
b. the set of elements in B but not A
c. the set of elements in U that are not in A
d. the set of elements in U that are not in A or B.
The Complement of a Set
In arithmetic, we use operations such as addition and multiplication to combine
numbers. We now turn to three set operations, called complement, intersection, and union. We begin by defining a set’s complement.
4 Find the complement of a set.
D E F I N I T I O N O F T H E C O M P L E M E N T O F A S E T
The complement of set A, symbolized by A′, is the set of all elements in the universal set that are not in A. This idea can be expressed in set-builder notation as follows:
A′ = 5x� x∊U and x∉A6.
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 77
The Intersection of Sets
If A and B are sets, we can form a new set consisting of all elements that are in both A and B. This set is called the intersection of the two sets.
The shaded region in Figure 2.13 represents the complement of set A, or A′. This region lies outside circle A, but within the rectangular universal set.
In order to find A′, a universal set U must be given. A fast way to find A′ is to cross out the elements in U that are given to be in set A. A′ is the set that remains.
A
A′
U
F I G U R E 2 . 1 3
A 1 3 4 7
2 5 6 8 9A′
U
F I G U R E 2 . 1 4
5 Find the intersection of two sets.
EXAMPLE 3 Finding a Set’s Complement
Let U = 51, 2, 3, 4, 5, 6, 7, 8, 96 and A = 51, 3, 4, 76. Find A′.
SOLUTION
Set A′ contains all the elements of set U that are not in set A. Because set A contains the elements 1, 3, 4, and 7, these elements cannot be members of set A′:
5 1 , 2, 3 , 4 , 5, 6, 7 , 8, 96.
Thus, set A′ contains 2, 5, 6, 8, and 9:
A′ = 52, 5, 6, 8, 96.
A Venn diagram illustrating A and A′ is shown in Figure 2.14.
CHECK POINT 3 Let U = 5a, b, c, d, e6 and A = 5a, d6. Find A′.
D E F I N I T I O N O F T H E I N T E R S E C T I O N O F S E T S
The intersection of sets A and B, written A ¨ B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows:
A ¨ B = 5x� x∊A and x∊B6.
In Example 4, we are asked to find the intersection of two sets. This is done by
listing the common elements of both sets. Because the intersection of two sets is
also a set, we enclose these elements with braces.
EXAMPLE 4 Finding the Intersection of Two Sets
Find each of the following intersections:
a. 57, 8, 9, 10, 116¨56, 8, 10, 126 b. 51, 3, 5, 7, 96¨52, 4, 6, 86 c. 51, 3, 5, 7, 96¨ ∅.
SOLUTION
a. The elements common to 57, 8, 9, 10, 116 and 56, 8, 10, 126 are 8 and 10. Thus,
57, 8, 9, 10, 116¨56, 8, 10, 126 = 58, 106.
The Venn diagram in Figure 2.15 illustrates this situation.
U
7 9 11
8 10
6 12
F I G U R E 2 . 1 5 The numbers 8 and 10
belong to both sets.
78 C H A P T E R 2 Set Theory
b. The sets 51, 3, 5, 7, 96 and 52, 4, 6, 86 have no elements in common. Thus,
51, 3, 5, 7, 96¨52, 4, 6, 86 = ∅.
The Venn diagram in Figure 2.16 illustrates this situation. The sets are disjoint.
c. There are no elements in ∅, the empty set. This means that there can be no elements belonging to both 51, 3, 5, 7, 96 and ∅. Therefore,
51, 3, 5, 7, 96¨ ∅ = ∅.
The Union of Sets
Another set that we can form from sets A and B consists of elements that are in A or B or in both sets. This set is called the union of the two sets.
U
1 3 5 7 9
2 4 6 8
F I G U R E 2 . 1 6 These disjoint sets have
no common elements.
CHECK POINT 4 Find each of the following intersections: a. 51, 3, 5, 7, 106¨56, 7, 10, 116 b. 51, 2, 36¨54, 5, 6, 76 c. 51, 2, 36¨ ∅.
GREAT QUESTION!
Set theory seems so abstract. For instance, how do I come across the intersection of two sets in my daily life?
Here’s an example: TV celebrities earning more than $80 million. This is the intersection
of the set of TV celebrities and the set of people earning more than $80 million. It’s easy
not to notice set theory, but if you look at the media and listen closely to conversations,
it’s all over the place.
TV Celebrities Earning More Than $80 Million between June 2013 and June 2014
Howard
Stern
Simon
Cowell
Glenn
Beck
Oprah
Winfrey
Dr. Phil
McGraw
20
E a rn
in g s
(m il
li o
n s
o f
d o
ll a rs
)
40
60
80
100
120
140
160
180
200
$82 million $82 million $90 million$95 million$95 million
Source: Forbes
6 Find the union of two sets.
D E F I N I T I O N O F T H E U N I O N O F S E T S
The union of sets A and B, written A ∪ B, is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set-builder notation as follows:
A ∪ B = 5x� x∊A or x∊B6.
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 79
We can find the union of set A and set B by listing the elements of set A. Then, we include any elements of set B that have not already been listed. Enclose all elements that are listed with braces. This shows that the union of two sets is
also a set.
EXAMPLE 5 Finding the Union of Two Sets
Find each of the following unions:
a. 57, 8, 9, 10, 116 ∪ 56, 8, 10, 126 b. 51, 3, 5, 7, 96 ∪ 52, 4, 6, 86 c. 51, 3, 5, 7, 96 ∪ ∅.
SOLUTION
This example uses the same sets as in Example 4. However, this time we are
finding the unions of the sets, rather than their intersections.
a. To find 57, 8, 9, 10, 116 ∪ 56, 8, 10, 126, start by listing all the elements from the first set, namely 7, 8, 9, 10, and 11. Now list all the elements
from the second set that are not in the first set, namely 6 and 12. The
union is the set consisting of all these elements. Thus,
57, 8, 9, 10, 116 ∪ 56, 8, 10, 126 = 56, 7, 8, 9, 10, 11, 126.
b. To find 51, 3, 5, 7, 96 ∪ 52, 4, 6, 86, list the elements from the first set, namely 1, 3, 5, 7, and 9. Now add to the list the elements in the second
set that are not in the first set. This includes every element in the second
set, namely 2, 4, 6, and 8. The union is the set consisting of all these
elements, so
51, 3, 5, 7, 96 ∪ 52, 4, 6, 86 = 51, 2, 3, 4, 5, 6, 7, 8, 96.
c. To find 51, 3, 5, 7, 96 ∪ ∅, list the elements from the first set, namely 1, 3, 5, 7, and 9. Because there are no elements in ∅, the empty set, there are no additional elements to add to the list. Thus,
51, 3, 5, 7, 96 ∪ ∅ = 51, 3, 5, 7, 96.
When finding the union of two sets, what should I do if some elements appear in both sets?
List these common elements
only once, not twice, in the union of the sets.
GREAT QUESTION!
Examples 4 and 5 illustrate the role that the empty set plays in intersection
and union.
T H E E M P T Y S E T I N I N T E R S E C T I O N A N D U N I O N
For any set A,
1. A ¨ ∅ = ∅ 2. A ∪ ∅ = A.
CHECK POINT 5 Find each of the following unions: a. 51, 3, 5, 7, 106 ∪ 56, 7, 10, 116 b. 51, 2, 36 ∪ 54, 5, 6, 76 c. 51, 2, 36 ∪ ∅.
80 C H A P T E R 2 Set Theory
Performing Set Operations
Some problems involve more than one set operation. The set notation specifies the order
in which we perform these operations. Always begin by performing any operations inside parentheses. Here are two examples involving sets we will find in Example 6.
• Finding (A ´ B)′
• Finding A′ ¨ B′
A B
A ´ B
A
B
A′ B′
7 Perform operations with sets.
How can I use the words union and intersection to help me distinguish between these two operations?
Union, as in a marriage union,
suggests joining things, or
uniting them. Intersection,
as in the intersection of two
crossing streets, brings to mind
the area common to both,
suggesting things that overlap.
GREAT QUESTION!
EXAMPLE 6 Performing Set Operations
Given
U = 51, 2, 3, 4, 5, 6, 7, 8, 9, 106
A = 51, 3, 7, 96
B = 53, 7, 8, 106,
find each of the following sets:
a. (A ∪ B)′ b. A′ ¨ B′.
SOLUTION
a. To find (A ∪ B)′, we will first work inside the parentheses and determine A ∪ B. Then we’ll find the complement of A ∪ B, namely (A ∪ B)′.
A ∪ B = 51, 3, 7, 96 ∪ 53, 7, 8, 106 These are the given sets. = 51, 3, 7, 8, 9, 106 Join (unite) the elements, listing the
common elements (3 and 7) only once.
Now find (A ∪ B)′, the complement of A ∪ B.
(A ∪ B)′ = 51, 3, 7, 8, 9, 106′ = 52, 4, 5, 66 List the elements in the universal set
that are not listed in 51, 3, 7, 8, 9, 106: 5 1 , 2, 3 , 4, 5, 6, 7 , 8 , 9 , 10 6.
b. To find A′ ¨ B′, we must first identify the elements in A′ and B′. Set A′ is the set of elements of U that are not in set A:
A′ = 52, 4, 5, 6, 8, 106. List the elements in the universal set that are not listed in A = 51, 3, 7, 96: 5 1 , 2, 3 , 4, 5, 6, 7 , 8, 9 , 106.
Set B′ is the set of elements of U that are not in set B:
B′ = 51, 2, 4, 5, 6, 96. List the elements in the universal set that are not listed in B = 53, 7, 8, 106: 51, 2, 3 , 4, 5, 6, 7 , 8 , 9, 10 6.
Now we can find A′ ¨ B′, the set of elements belonging to both A′ and to B′:
A′ ¨ B′ = 52, 4, 5, 6, 8, 106¨51, 2, 4, 5, 6, 96 = 52, 4, 5, 66. The numbers 2, 4, 5, and 6
are common to both sets.
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 81
Set to Determine Description of Set Regions in Venn
Diagram in Figure 2.17 Set in Roster Form
a. A ∪ B set of elements in A or B or both I, II, III 5p, e, 22 , 2- 1 , epi, 10100, 2ℵ06
b. (A ∪ B)′ set of elements in U that are not in A ∪ B IV 56666
c. A ¨ B set of elements in both A and B II 522 , 2- 16
d. (A ¨ B)′ set of elements in U that are not in A ¨ B I, III, IV 5p, e, epi, 10100, 2ℵ0, 6666
e. A′ ¨ B set of elements that are not in A and are in B III 5epi, 10100, 2ℵ06
f. A ∪ B′ set of elements that are in A or not in B or both I, II, IV 5p, e, 22 , 2- 1 , 6666
CHECK POINT 6 Given U = 5a, b, c, d, e6, A = 5b, c6, and B = 5b, c, e6, find each of the following sets:
a. (A ∪ B)′ b. A′ ¨ B′.
8 Determine sets involving set operations from a Venn diagram. II
A U
B III
IV 666
I p e
epi
10100
2u 0
"2 "-1
F I G U R E 2 . 1 7
EXAMPLE 7 Determining Sets from a Venn Diagram
The Venn diagram in Figure 2.17 percolates with interesting numbers. Use the diagram
to determine each of the following sets:
a. A ∪ B b. (A ∪ B)′ c. A ¨ B d. (A ¨ B)′ e. A′ ¨ B f. A ∪ B′.
SOLUTION
Refer to Figure 2.17.
CHECK POINT 7 Use the Venn diagram in Figure 2.18 to determine each of the following sets:
a. A ¨ B b. (A ¨ B)′ c. A ∪ B d. (A ∪ B)′ e. A′ ∪ B f. A ¨ B′.
II A
U B
III
IV 17 19
I 2 3
5 7 11 13
F I G U R E 2 . 1 8
9 Understand the meaning of andand or. Sets and Precise Use of Everyday English Set operations and Venn diagrams provide precise ways of organizing, classifying,
and describing the vast array of sets and subsets we encounter every day. Let’s see
how this applies to the sets from the beginning of this section:
U = the set of U.S. Hispanics S = the set of U.S. Hispanics who speak Spanish at home E = the set of U.S. Hispanics who speak English at home.
82 C H A P T E R 2 Set Theory
When describing collections in everyday English, the word or refers to the union of sets. Thus, U.S. Hispanics who speak Spanish or English at home means those who speak Spanish or English or both. The word and refers to the intersection of sets. Thus, U.S. Hispanics who speak Spanish and English at home means those
who speak both languages.
In Figure 2.19, we revisit the circle graph showing languages spoken at home by U.S. Hispanics. To the right of the circle graph, we’ve organized the data using a
Venn diagram. The voice balloons indicate how the Venn diagram provides a more
accurate understanding of the subsets and their data.
Languages Spoken at Home by U.S. Hispanics
English 32%
Both 23%
Spanish 43%
Other 2%
II
U
III
2% IV
I
43% 23% 32%
Sp anish English
+ + =
S ´ E S ¨ E
+ =+ =
S ¨ E′ E ¨ S′
F I G U R E 2 . 1 9 Comparing a circle graph and a Venn diagram
Source: Time
The Cardinal Number of the Union of Two Finite Sets
Can the number of elements in A or B, n(A ∪ B), be determined by adding the number of
elements in A and the number of elements in B, n(A) + n(B)? The answer is no. Figure 2.20 illustrates that by doing this, we are counting
elements in both sets, A ¨ B, or region II, twice. To find the number of elements in the
union of finite sets A and B, add the number of elements in A and the number of elements in B. Then subtract the number of elements common
to both sets. We perform this subtraction so
that we do not count the number of elements in
the intersection twice, once for n(A), and again for n(B).
10 Use the formula for n1A ∪ B2. U
A B
I II
IV
III
n A
I II
n B
II III
F I G U R E 2 . 2 0
F O R M U L A F O R T H E C A R D I N A L N U M B E R O F T H E U N I O N
O F T W O F I N I T E S E T S
n(A ´ B) = n(A) + n(B) - n(A ¨ B)
A B A
B A B
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 83
EXAMPLE 8 Using the Formula for n1A ∪ B2
Some of the results of the campus blood drive survey indicated that
490 students were willing to donate blood, 340 students were willing to help
serve a free breakfast to blood donors, and 120 students were willing to donate
blood and serve breakfast. How many students were willing to donate blood
or serve breakfast?
SOLUTION
Let A = the set of students willing to donate blood and B = the set of students willing to serve breakfast. We are interested in how many students were willing
to donate blood or serve breakfast. Thus, we need to determine n(A ∪ B).
n(A ´ B) = n(A) + n(B) - n(A ¨ B)
= 490 + 340 - 120
= 830 - 120
= 710
We see that 710 students were willing to donate blood or serve a free breakfast.
CHECK POINT 8 According to factmonster.com, among the U.S. presidents in the White House, 26 had dogs, 11 had cats, and 9 had both dogs and cats. How
many U.S. presidents had dogs or cats in the White House?
Presidents with Dogs
Presidents with Cats
Fill in each blank so that the resulting statement is true.
1. Visual relationships among sets are shown by ________________.
2. The set of all elements in the universal set that are not in set A is called the ______________ of set A, and is symbolized by _____.
3. The set of elements common to both set A and set B is called the _____________ of sets A and B, and is symbolized by _________.
4. The set of elements that are members of set A or set B or of both sets is called the ________ of sets A and B, and is symbolized by _________.
5. The formula for the cardinal number of elements in set A or set B is n 1A ∪ B2 = ____________________________.
6. True or False: Disjoint sets are represented by circles that do not overlap. _______
7. True or False: If set A is a proper subset of set B, the sets are represented by two circles where circle A is drawn outside of circle B. _______
8. True or False: Equal sets are represented by the same circle. _______
9. True or False: As the number of elements in a set increases, larger circles are needed to represent the set. _______
Concept and Vocabulary Check
84 C H A P T E R 2 Set Theory
Practice Exercises
In Exercises 1–4, describe a universal set U that includes all elements in the given sets. Answers may vary.
1. A = 5Bach, Mozart, Beethoven6 B = 5Brahms, Schubert6
2. A = 5William Shakespeare, Charles Dickens6 B = 5Mark Twain, Robert Louis Stevenson6
3. A = 5Pepsi, Sprite6 B = 5Coca@Cola, Seven@Up6
4. A = 5Acura RDX, Toyota Camry, Mitsubishi Lancer6 B = 5Dodge Ram, Chevrolet Impala6
In Exercises 5–8, let U = 5a, b, c, d, e, f, g6, A = 5a, b, f, g6, B = 5c, d, e6, C = 5a, g6, and D = 5a, b, c, d, e, f6. Use the roster method to write each of the following sets.
5. A′ 6. B′ 7. C′ 8. D′
In Exercises 9–12, let U = 51, 2, 3, 4, c, 206, A = 51, 2, 3, 4, 56, B = 56, 7, 8, 96, C = 51, 3, 5, 7, c, 196, and D = 52, 4, 6, 8, c, 206. Use the roster method to write each of the following sets.
9. A′ 10. B′ 11. C′ 12. D′
In Exercises 13–16, let U = 51, 2, 3, 4, c6, A = 51, 2, 3, 4, c, 206, B = 51, 2, 3, 4, c, 506, C = 52, 4, 6, 8, c6, and D = 51, 3, 5, 7, c6. Use the roster method to write each of the following sets.
13. A′ 14. B′ 15. C′ 16. D′
In Exercises 17–40, let
U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, 36 C = 52, 3, 4, 5, 66.
Find each of the following sets.
17. A ¨ B 18. B ¨ C 19. A ∪ B 20. B ∪ C 21. A′ 22. B′ 23. A′ ¨ B′ 24. B′ ¨ C 25. A ∪ C′ 26. B ∪ C′ 27. (A ¨ C)′ 28. (A ¨ B)′ 29. A′ ∪ C′ 30. A′ ∪ B′ 31. (A ∪ B)′ 32. (A ∪ C)′ 33. A ∪ ∅ 34. C ∪ ∅ 35. A ¨ ∅ 36. C ¨ ∅ 37. A ∪ U 38. B ∪ U 39. A ¨ U 40. B ¨ U In Exercises 41–66, let
U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, h6 C = 5b, c, d, e, f6.
Find each of the following sets.
41. A ¨ B 42. B ¨ C 43. A ∪ B 44. B ∪ C 45. A′ 46. B′ 47. A′ ¨ B′ 48. B′ ¨ C 49. A ∪ C′ 50. B ∪ C′ 51. (A ¨ C)′ 52. (A ¨ B)′ 53. A′ ∪ C′ 54. A′ ∪ B′ 55. (A ∪ B)′ 56. (A ∪ C)′ 57. A ∪ ∅ 58. C ∪ ∅ 59. A ¨ ∅ 60. C ¨ ∅ 61. A ∪ U 62. B ∪ U 63. A ¨ U 64. B ¨ U 65. (A ¨ B) ∪ B′ 66. (A ∪ B) ¨ B′
In Exercises 67–78, use the Venn diagram to represent each set in roster form.
U A B
1 4
3 7
8 9
2 5 6
67. A 68. B
69. U 70. A ∪ B 71. A ¨ B 72. A′ 73. B′ 74. (A ¨ B)′ 75. (A ∪ B)′ 76. A′ ¨ B 77. A ¨ B′ 78. A ∪ B′
In Exercises 79–92, use the Venn diagram to determine each set or cardinality.
A U
B
10 01
# $
Δ two six
four
79. B 80. A 81. A ∪ B 82. A ¨ B 83. n(A ∪ B) 84. n(A ¨ B) 85. n(A′) 86. n(B′) 87. (A ¨ B)′ 88. (A ∪ B)′ 89. A′ ¨ B 90. A ¨ B′ 91. n(U) - n(B) 92. n(U) - n(A)
Use the formula for the cardinal number of the union of two sets to solve Exercises 93–96.
93. Set A contains 17 elements, set B contains 20 elements, and 6 elements are common to sets A and B. How many elements are in A ∪ B?
94. Set A contains 30 elements, set B contains 18 elements, and 5 elements are common to sets A and B. How many elements are in A ∪ B?
Exercise Set 2.3
S E C T I O N 2 . 3 Venn Diagrams and Set Operations 85
95. Set A contains 8 letters and 9 numbers. Set B contains 7 letters and 10 numbers. Four letters and 3 numbers are
common to both sets A and B. Find the number of elements in set A or set B.
96. Set A contains 12 numbers and 18 letters. Set B contains 14 numbers and 10 letters. One number and 6 letters are
common to both sets A and B. Find the number of elements in set A or set B.
Practice Plus
In Exercises 97–104, let
U = 5x� x∊N and x 6 96 A = 5x� x is an odd natural number and x 6 96 B = 5x� x is an even natural number and x 6 96 C = 5x� x∊N and 1 6 x 6 66.
Find each of the following sets.
97. A ∪ B 98. B ∪ C 99. A ¨ U 100. A ∪ U 101. A ¨ C′ 102. A ¨ B′ 103. (B ¨ C)′ 104. (A ¨ C)′
In Exercises 105–108, use the Venn diagram to determine each set or cardinality.
A U
B
53 59 61 67 71
23 29 31 37
41 43
47
105. A ∪ (A ∪ B)′ 106. (A′ ¨ B) ∪ (A ¨ B) 107. n(U)[n(A ∪ B) - n(A ¨ B)] 108. n(A ¨ B)[n(A ∪ B) - n(A′)]
Application Exercises
A math tutor working with a small group of students asked each student when he or she had studied for class the previous weekend. Their responses are shown in the Venn diagram.
Jacob
Studied Saturday
Studied Sunday
Ashley Mike Josh
Emily Hanna
Ethan
U
In Exercises 109–116, use the Venn diagram to list the elements of each set in roster form.
109. The set of students who studied Saturday
110. The set of students who studied Sunday
111. The set of students who studied Saturday or Sunday
112. The set of students who studied Saturday and Sunday
113. The set of students who studied Saturday and not Sunday
114. The set of students who studied Sunday and not Saturday
115. The set of students who studied neither Saturday nor Sunday
116. The set of students surveyed by the math tutor
The bar graph shows the percentage of Americans with gender preferences for various jobs.
70%
60%
50%
40%
30%
20%
P e rc
e n
ta g e o
f A
m e ri
c a n s
Gender and Jobs: Percentage of Americans Who Prefer Men or Women in Various Jobs
LawyerFamily Doctor
Airline Pilot
SurgeonBankerPolice Officer
Elementary School Teacher
10%
Prefer men Prefer women No Preference
Source: Pew Research Center
In Exercises 117–122, use the information in the graph to place the indicated job in the correct region of the following Venn diagram.
U A B
I II
IV
III
U =
A = B =
117. elementary school teacher 118. police officer
119. surgeon 120. banker
121. family doctor 122. lawyer
A palindromic number is a natural number whose value does not change if its digits are reversed. Examples of palindromic numbers are 11, 454, and 261,162. In Exercises 123–132, use this definition to place the indicated natural number in the correct region of the following Venn diagram.
U A B
I II
IV
III
U =
A = B =
123. 11 124. 22 125. 15 126. 17
127. 454 128. 101 129. 9558 130. 9778
131. 9559 132. 9779
86 C H A P T E R 2 Set Theory
The bar graph shows the percentage of Americans, by age group, supporting legalized marijuana for four selected years from 1969 through 2015. Use the information in the graph to write each set in Exercises 133–138 in roster form or express the set as ∅.
80%
70%
60%
50%
30%
P e rc
e n
ta g e S
u p
p o
rt in
g L
e g a li
z e d
A d
u lt
M a ri
ju a n
a U
se
Generational Support for Legalizing Adult Marijuana Use
35
58 65
71
15
28 32
42
12 18
22
31
1969 1985 2001 2015
2 5
11
20
10%
40%
20%
18–34
Age Groups
35–49
Year
50–64 65+
Source: USA TODAY
133. 5x� x was a year in which more than 40% of age group 18–34 supported legalization6 ¨ 5x� x was a year in which fewer than 20% of age group 65 + supported legalization6
134. 5x� x was a year in which more than 30% of age group 18–34 supported legalization6 ¨ 5x� x was a year in which fewer than 14% of age group 65 + supported legalization6
135. 5x� x was a year in which more than 40% of age group 18–34 supported legalization6 ∪ 5x� x was a year in which fewer than 20% of age group 65 + supported legalization6
136. 5x� x was a year in which more than 30% of age group 18–34 supported legalization6 ∪ 5x� x was a year in which fewer than 14% of age group 65 + supported legalization6
137. The set of years in which more than 50% of age group 18–34 supported legalization and fewer than 35% of age
group 35–49 supported legalization
138. The set of years in which more than 50% of age group 18–34 supported legalization or fewer than 35% of age
group 35–49 supported legalization
139. A winter resort took a poll of its 350 visitors to see which winter activities people enjoyed. The results were as follows:
178 people liked to ski, 154 people liked to snowboard, and
49 people liked to ski and snowboard. How many people in
the poll liked to ski or snowboard?
140. A pet store surveyed 200 pet owners and obtained the following results: 96 people owned cats, 97 people owned
dogs, and 29 people owned cats and dogs. How many
people in the survey owned cats or dogs?
Explaining the Concepts
141. Describe what is meant by a universal set. Provide an example.
142. What is a Venn diagram and how is it used?
143. Describe the Venn diagram for two disjoint sets. How does this diagram illustrate that the sets have no common
elements?
144. Describe the Venn diagram for proper subsets. How does this diagram illustrate that the elements of one set are also
in the second set?
145. Describe the Venn diagram for two equal sets. How does this diagram illustrate that the sets are equal?
146. Describe the Venn diagram for two sets with common elements. How does the diagram illustrate this relationship?
147. Describe what is meant by the complement of a set.
148. Is it possible to find a set’s complement if a universal set is not given? Explain your answer.
149. Describe what is meant by the intersection of two sets. Give an example.
150. Describe what is meant by the union of two sets. Give an example.
151. Describe how to find the cardinal number of the union of two finite sets.
Critical Thinking Exercises
Make Sense? In Exercises 152–155, determine whether each statement makes sense or does not make sense, and explain your reasoning.
152. Set A and set B share only one element, so I don’t need to use overlapping circles to visualize their relationship.
153. Even if I’m not sure how mathematicians define irrational and complex numbers, telling me how these sets are
related, I can construct a Venn diagram illustrating their
relationship.
154. If I am given sets A and B, the set (A ∪ B)′ indicates I should take the union of the complement of A and the complement of B.
155. I suspect that at least 90% of college students have no preference whether their professor is a man or a woman, so
I should place college professors in region IV of the Venn
diagram that precedes Exercises 117–122.
In Exercises 156–163, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
156. n(A ∪ B) = n(A) + n(B) 157. A ¨ A′ = ∅ 158. (A ∪ B) ⊆ A
159. If A ⊆ B, then A ¨ B = B. 160. A ¨ U = U 161. A ∪ ∅ = ∅
162. If A ⊆ B, then A ¨ B = ∅. 163. If B ⊆ A, then A ¨ B = B.
In Exercises 164–167, assume A ≠ B. Draw a Venn diagram that correctly illustrates the relationship between the sets.
164. A ¨ B = A 165. A ¨ B = B 166. A ∪ B = A
167. A ∪ B = B
S E C T I O N 2 . 4 Set Operations and Venn Diagrams with Three Sets 87
Set Operations and Venn Diagrams with Three Sets2.4
WHAT AM I SUPPOSED TO LEARN?
After studying this section, you
should be able to:
1 Perform set operations with three sets.
2 Use Venn diagrams with three sets.
3 Use Venn diagrams to prove equality of sets.
SHOULD YOUR BLOOD TYPE
determine what you eat? The
blood-type diet, developed by
naturopathic physician Peter
D’Adamo, is based on the
theory that people with
different blood types require
different diets for optimal
health. D’Adamo gives very
detailed recommendations
for what people with each
type should and shouldn’t eat.
For example, he says shitake
mushrooms are great for type B’s,
but bad for type O’s. Type B? Type O? In this section, we present a Venn diagram
with three sets that will give you a unique perspective on the different types of
human blood. Despite this perspective, we’ll have nothing to say about shitakes,
avoiding the question as to whether or not the blood-type diet really works.
Set Operations with Three Sets
We now know how to find the union and intersection of two sets. We also know
how to find a set’s complement. In Example 1, we apply set operations to situations
containing three sets.
1 Perform set operations with three sets.
EXAMPLE 1 Set Operations with Three Sets
Given U = 51, 2, 3, 4, 5, 6, 7, 8, 96
A = 51, 2, 3, 4, 56
B = 51, 2, 3, 6, 86
C = 52, 3, 4, 6, 76,
find each of the following sets:
a. A ∪ (B ¨ C) b. (A ∪ B) ¨ (A ∪ C) c. A ¨ (B ∪ C′).
SOLUTION
Before determining each set, let’s be sure we perform the operations in the
correct order. Remember that we begin by performing any set operations
inside parentheses.
• Finding A ´ (B ¨ C)
A B ¨ C
B C
• Finding (A ´ B) ¨ (A ´ C)
A B
A ´ B A ´ C
A C
88 C H A P T E R 2 Set Theory
• Finding A ¨ (B ´ C′) C
B C′
A B ´ C′
a. To find A ∪ (B ¨ C), first find the set within the parentheses, B ¨ C:
B ¨ C = 51, 2, 3, 6, 86 ¨ 52, 3, 4, 6, 76 = 52, 3, 66.
Now finish the problem by finding A ∪ (B ¨ C):
A ´ (B ¨ C) = 51, 2, 3, 4, 56 ´ 52, 3, 66 = 51, 2, 3, 4, 5, 66.
A B ¨ C
b. To find (A ∪ B) ¨ (A ∪ C), first find the sets within parentheses. Start with A ∪ B:
A ´ B = 51, 2, 3, 4, 56 ´ 51, 2, 3, 6, 86 = 51, 2, 3, 4, 5, 6, 86.
A B
Now find A ∪ C:
A ´ C = 51, 2, 3, 4, 56 ´ 52, 3, 4, 6, 76 = 51, 2, 3, 4, 5, 6, 76.
A C
Now finish the problem by finding (A ∪ B) ¨ (A ∪ C):
(A ´ B) ¨ (A ´ C) = 51, 2, 3, 4, 5, 6, 86 ¨ 51, 2, 3, 4, 5, 6, 76 = 51, 2, 3, 4, 5, 66.
c. As in parts (a) and (b), to find A ¨ (B ∪ C′), begin with the set in parentheses. First we must find C′, the set of elements in U that are not in C:
C′ = 51, 5, 8, 96. List the elements in U that are not in C = 52, 3, 4, 6, 76: 51, 2 , 3 , 4 , 5, 6 , 7 , 8, 96.
Now we can identify elements of B ∪ C′:
B ´ C′ = 51, 2, 3, 6, 86 ´ 51, 5, 8, 96 = 51, 2, 3, 5, 6, 8, 96.
B C′
Now finish the problem by finding A ¨ (B ∪ C′):
A ¨ (B ´ C′) = 51, 2, 3, 4, 56 ¨ 51, 2, 3, 5, 6, 8, 96 = 51, 2, 3, 56.
U = 51, 2, 3, 4, 5, 6, 7, 8, 96 A = 51, 2, 3, 4, 56 B = 51, 2, 3, 6, 86 C = 52, 3, 4, 6, 76,
The given sets (repeated)
S E C T I O N 2 . 4 Set Operations and Venn Diagrams with Three Sets 89
Venn Diagrams with Three Sets
Venn diagrams can contain three or more sets, such as the diagram in Figure 2.21. The three sets in the figure separate the universal set, U, into eight regions. The numbering of these regions is arbitrary—that is, we can number any region as I,
any region as II, and so on. Here is a description of each region, starting with the
innermost region, region V, and working outward to region VIII.
The Region Shown in Dark Blue
Region V This region represents elements that are common to sets A, B, and C: A ¨ B ¨ C.
The Regions Shown in Light Blue
Region II This region represents elements in both sets A and B that are not in set C: (A ¨ B) ¨ C′.
Region IV This region represents elements in both sets A and C that are not in set B: (A ¨ C) ¨ B′.
Region VI This region represents elements in both sets B and C that are not in set A: (B ¨ C) ¨ A′.
The Regions Shown in White
Region I This region represents elements in set A that are in neither sets B nor C: A ¨ (B′ ¨ C′).
Region III This region represents elements in set B that are in neither sets A nor C: B ¨ (A′ ¨ C′).
Region VII This region represents elements in set C that are in neither sets A nor B: C ¨ (A′ ¨ B′).
Region VIII This region represents elements in the universal set U that are not in sets A, B, or C: A′ ¨ B′ ¨ C′.
CHECK POINT 1 Given U = 5a, b, c, d, e, f6, A = 5a, b, c, d6, B = 5a, b, d, f6, and C = 5b, c, f6, find each of the following sets:
a. A ∪ (B ¨ C) b. (A ∪ B) ¨ (A ∪ C) c. A ¨ (B ∪ C′).
2 Use Venn diagrams with three sets.
I
A U
B
C
II III
IV
V
VI
VII
VIII
F I G U R E 2 . 2 1 Three intersecting sets
separate the universal set into eight
regions.
EXAMPLE 2 Determining Sets from a Venn Diagram with Three Intersecting Sets
Use the Venn diagram in Figure 2.22 to determine each of the following sets:
a. A b. A ∪ B c. B ¨ C d. C′ e. A ¨ B ¨ C.
I 11 3
A U
B
C
II 12
III 1 2 10
IV 6
V 5 7
VI 9
VII 8
4 VIII
F I G U R E 2 . 2 2
90 C H A P T E R 2 Set Theory
CHECK POINT 2 Use the Venn diagram in Figure 2.22 to determine each of the following sets:
a. C b. B ∪ C c. A ¨ C d. B′ e. A ∪ B ∪ C.
SOLUTION
Set to Determine Description of Set Regions in Venn
Diagram Set in Roster Form
a. A set of elements in A I, II, IV, V 511, 3, 12, 6, 5, 76
b. A ∪ B set of elements in A or B or both I, II, III, IV, V, VI 511, 3, 12, 1, 2, 10, 6, 5, 7, 96
c. B ¨ C set of elements in both B and C V, VI 55, 7, 96
d. C′ set of elements in U that are not in C I, II, III, VIII 511, 3, 12, 1, 2, 10, 46
e. A ¨ B ¨ C set of elements in A and B and C V 55, 76
In Example 2, we used a Venn diagram showing elements in the regions to
determine various sets. Now we are going to reverse directions. We’ll use sets
A, B, C, and U to determine the elements in each region of a Venn diagram. To construct a Venn diagram illustrating the elements in A, B, C, and U,
start by placing elements into the innermost region and work outward. Because the four inner regions represent various intersections, find A ¨ B, A ¨ C, B ¨ C, and A ¨ B ¨ C. Then use these intersections and the given sets to place the various elements into regions. This procedure is illustrated in Example 3.
EXAMPLE 3 Determining a Venn Diagram from Sets
Construct a Venn diagram illustrating the following sets:
A = 5a, d, e, g, h, i, j6 B = 5b, e, g, h, l6 C = 5a, c, e, h6 U = 5a, b, c, d, e, f, g, h, i, j, k, l6.
SOLUTION
We begin by finding four intersections. In each case, common elements are
shown in red.
• A ¨ B ¨ C = 5e, g, h6 ¨ 5a, c, e, h6 = 5e, h6 • B ¨ C = 5b, e, g, h, l6 ¨ 5a, c, e, h6 = 5e, h6 • A ¨ C = 5a, d, e, g, h, i, j6 ¨ 5a, c, e, h6 = 5a, e, h6 • A ¨ B = 5a, d, e, g, h, i, j6 ¨ 5b, e, g, h, l6 = 5e, g, h6
A ¨ B
Now we can place elements into regions, starting with the innermost region,
region V, and working outward.
I 11 3
A U
B
C
II 12
III 1 2 10
IV 6
V 5 7
VI 9
VII 8
4 VIII
F I G U R E 2 . 2 2 (repeated)
S E C T I O N 2 . 4 Set Operations and Venn Diagrams with Three Sets 91
Before placing elements into regions, let’s repeat the four intersections that we found:
A ¨ B ¨ C = 5e, h6, A ¨ B = 5e, g, h6, A ¨ C = 5a, e, h6, B ¨ C = 5e, h6.
STEP 1
I
A U
B
C
II III
IV
V e h VI
VII
VIII
A ¨ B ¨ C V
A ¨ B ¨ C = V
STEP 2
I
A U
B
C
II III
IV
V e h
g
VI
VII
VIII
A ¨ B II V
A ¨ B = V,
II
STEP 3
I
A U
B
C
II III
IV
V e h
g
a VI
VII
VIII
A ¨ C IV V
A ¨ C = V
IV
STEP 4
I
A U
B
C
II III
IV
V e h
g
a VI
VII
VIII
B ¨ C V VI
B ¨ C = V, VI
STEP 5
I d i j
A U
B
C
II III
IV
V e h
g
a VI
VII
VIII
A I II IV V
A =
A I
STEP 6
I d i j
A U
B
C
II III
IV
V e h
g b l
a VI
VII
VIII
B II III V VI
B =
B III
STEP 7
I d i j
A U
B
C
II III
IV
V e h
g
c
b l
a VI
VII
VIII
C IV V VI VII
C =
C VII
STEP 8
I d i j
A U
B
C
II III
IV
V e h
f k
g
c
b l
a VI
VII
VIII
U I–VIII
U =
U VIII
The completed Venn diagram in step 8 illustrates the given sets.
CHECK POINT 3 Construct a Venn diagram illustrating the following sets: A = 51, 3, 6, 106 B = 54, 7, 9, 106 C = 53, 4, 5, 8, 9, 106 U = 51, 2, 3, 4, 5, 6, 7, 8, 9, 106.
3 Use Venn diagrams to prove equality of sets. Proving the Equality of Sets Throughout Section 2.3, you were given two sets A and B and their universal set U and asked to find (A ¨ B)′ and A′ ∪ B′. In each example, (A ¨ B)′ and A′ ∪ B′ resulted in the same set. This occurs regardless of which sets we choose for A and B in a universal set U. Examining these individual cases and applying inductive reasoning, a conjecture (or educated guess) is that (A ¨ B)′ = A′ ∪ B′.
92 C H A P T E R 2 Set Theory
We can apply deductive reasoning to prove the statement (A ¨ B)′ = A′ ∪ B′ for all sets A and B in any universal set U. To prove that (A ¨ B)′ and A′ ∪ B′ are equal, we use a Venn diagram. If both sets are represented by the same regions in
this general diagram, then this proves that they are equal. Example 4 shows how
this is done.
A BRIEF REVIEW In summary, here are the two
forms of reasoning discussed
in Chapter 1.
• Inductive Reasoning: Starts with individual
observations and works to
a general conjecture (or
educated guess)
• Deductive Reasoning: Starts with general cases
and works to the proof of
a specific statement (or
theorem)
U A B
I II
IV
III
F I G U R E 2 . 2 3
EXAMPLE 4 Proving the Equality of Sets
Use the Venn diagram in Figure 2.23 to prove that
(A ¨ B)′ = A′ ∪ B′.
SOLUTION
Begin by identifying the regions representing (A ¨ B)′.
Set Regions in the Venn Diagram
A I, II
B II, III
A ¨ B II (This is the region common to A and B.)
(A ¨ B)′ I, III, IV (These are the regions in U that are not in A ¨ B.)
Next, find the regions in Figure 2.23 representing A′ ∪ B′.
Set Regions in the Venn Diagram
A′ III, IV (These are the regions not in A.)
B′ I, IV (These are the regions not in B.)
A′ ∪ B′ I, III, IV (These are the regions obtained by uniting the regions representing A′ and B′.)
Both (A ¨ B)′ and A′ ∪ B′ are represented by the same regions, I, III, and IV, of the Venn diagram. This result proves that
(A ¨ B)′ = A′ ∪ B′
for all sets A and B in any universal set U.
Can you see how we applied deductive reasoning in Example 4? We started
with the two general sets in the Venn diagram in Figure 2.23 and worked to the specific conclusion that (A ¨ B)′ and A′ ∪ B′ represent the same regions in the diagram. Thus, the statement (A ¨ B)′ = A′ ∪ B′ is a theorem.
CHECK POINT 4 Use the Venn diagram in Figure 2.23 to solve this exercise. a. Which region represents (A ∪ B)′? b. Which region represents A′ ¨ B′? c. Based on parts (a) and (b), what can you conclude?