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logic.docDiscrete Structures
Readings Check section 3.1
Read Section 3.1, pages 123 ( 134, and answer the questions below:.
[Note: you do not need to read examples 3.9, 3.10, or 3.11]
Type in the answers below each question and email the completed document to me, or print out the document and fill it out by hand and email a scan or photo of it to me.
1) Does the universal set U contain everything in the universe? What does it (or can it) contain?
2) What is the cardinality of a set? What notation is used for cardinality?
3) What is the difference between saying that A ( B, and saying that A ( B ? Be precise in your answer.
4) If C ( D and D ( C, what else can we say about the relationship between C and D?
5) If A ( B and B ( C, what is the relationship between A and C?
6) Is it true or false that ( = {(} ? [see definition 3.3]
7) In example 3.7 the notation
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calculate?8) What does the notation P(A) (the power set of A) represent?
9) Referring to example 3.14, If
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, then what are the values of n and k?Discrete Structures
Readings Check section 3.2
Read Section 3.2, pages 136 ( 144, except for the proofs of the theorems, and answer the questions below:.
Type in the answers below each question and email the completed document to me, or print out the document and fill it out by hand and email a scan or photo of it to me.
1) Which logical connective is associated with ( ?
2) Which logical connective is associated with ( ?
3) What does it mean for sets S and T to be disjoint?
4) What is meant by the notation
A
?5) Suppose that A and B are disjoint. Then what is A ( B, written as a single letter?
6) Compare the Laws of Set Theory on page 139 with the Laws of Logic on pages 58 ( 59. Write down several similarities between these two sets of laws.
7) Comparing the dual for logic to the dual for sets, what two sets assume the roles of T and F when it comes to creating the dual for sets?
8) A membership table very much resembles a truth table. In a truth table, the number 0 signifies false and the number 1 signifies true. With regard to a particular element x, what do the numbers 0 and 1 indicate in a membership table? [see page 143]
Discrete Structures
Readings Check section 3.3
Read Section 3.3, pages 148 ( 150.
Type in the answers below each question and email the completed document to me, or print out the document and fill it out by hand and email a scan or photo of it to me.
1) What is the relationship between A and B that needs to be true in order that |A ( B| = |A| + |B| ?
2) What is |A| + |B| ( |A ( B| always equal to?
3) In example 3.26, what is the numerical value of |
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| ?4) With regard to example 3.26, give a verbal description of the set
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in terms of gates and defects5) In example 3.26, what is the numerical value of |A ( B| ?
6) With regard to example 3.26, give a verbal description of the set A ( B in terms of gates and defects
7) What is |A| + |B| + |C| ( |A ( B| ( |A ( C| ( |B ( C| + |A ( B ( C| always equal to?
Discrete Structures
Readings Check section 3.4
Read Section 3.4, pages 150 ( 155.
Type in the answers below each question and email the completed document to me, or print out the document and fill it out by hand and email a scan or photo of it to me.
1) What does the sample space contain?
2) What is an event?
3) From example 3.29, what is a formula that relates Pr(A) to Pr(
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) ?4) Looking at example 3.32, what are the ordered pairs in the set B ( B ?
5) Using the sample space of two rolled dice in example 3.33 (also in the notes for this section), what is the probability that the product of the two numbers is evenly divisible by 6?
6) In example 3.35, what is the probability that he gets 1 head and 3 tails?
7) Using the data and the diagram from example 3.37, what is the probability that both passengers enjoyed only mixed drinks?
Weekly Summary, Week 3, Chapter 3, Discrete Structures Name:
Due on Sunday, February 5, by midnight.
Type your answers under the questions given below, or, print out this sheet, use pen or pencil to fill it out, and email a scan or photo of it to the instructor.
1) Explain Russell's Paradox, which is outlined in section 3.1, problem #27. What I am looking for here is a basic explanation, of around 5 ( 10 sentences, that is accessible to any student who understands section 3.1.
2) Suppose we look at the list of binomial coefficients:
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.a) If n is an even number, which binomial coefficient is the largest?
b) If n is an odd number, which two binomial coefficients are the largest? [Hint: take a look at Pascal's triangle.]
3) Here we will add to the list of the Laws of Set Theory. In the most reduced form possible, what do each of the following sets equal?
a) A ( A =
b) A (
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d) A ( U =
e) A (
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=f) If A ( B then A ( B =
4) Let S be a set with |S| = 10. Let the elements of set S1 be all two element subsets of S. Let the elements of set S2 be all two element subsets of S1. Which is larger, |P(S)| or |S2| ?
[Note: P(S) is the power set of the set S.]
5) If four dice (of different colors) are rolled, what is the probability that their sum equals 6? (ask for a hint)
bonus: Suppose that A ( B, B and C are disjoint, |B ( A| = 1502, |A| + |C| = 510, |
C
| = 1606, and |B
| = 414. What is |U| ?
