Logical Equivalencies and Direct Proof
Math 301 Summer 2020 Dr. Kimberly Vincent
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Exam I-Logical Equivalencies and Direct Proof
Write your solutions with only one question per page side of a sheet of paper. You may use both sides. (I will not read any work on this question sheet).
Follow all directions.
Justify all your work with good pictures, analytical work, explanations or a combination.
USE COMPLETE SENTENCES when explaining and writing your proofs so you complete your thoughts.
When you are done with your exam sign the following
statement and include in the pdf you upload to blackboard
with your responses.
Sign and include the following statement in the pdf of your
exam:
I acknowledge that all the work submitted for
exam 1 in Math 301, summer 2020 is my own
original work.
I acknowledge that I did not share my work or
copy anyone else’s work as part of this exam.
I further acknowledge that if I shared my
answers or copied other’s work we both get
zeros on the exam.
Printed Name__________________________
Signature_____________________________
Date_______________________________
Math 301 Summer 2020 Dr. Kimberly Vincent
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1. a. (5 pts.) Write in English a translation of the following symbols QPQP
b. (10 pts) Use truth tables to verify that QPQP
c. (5 pts.) Why does the truth table verify QPQP .
TURN TO NEW PAGE
2. (10 pts.) Use previously proven logical equivalencies, see Theorem 2.9 above, to prove the following:
(𝑷 → 𝑸) → 𝑹 ≡ (~𝑷 → 𝑹) ∧ (𝑸 → 𝑹) TURN TO NEW PAGE
3. Assume the universal set is the set of integers. )0)()(( nmnm
a. (5 pts) Write the statement above using an English sentence that does not use the symbols for quantifiers.
b. (10 pts.) Write the negation of the above statement in English and in symbolic form. c. (5 pts) Which statement is false? Provide a counter example.
TURN TO NEW PAGE
4. (15 pts.) Prove the following: If x is an even integer then x2 is an even integer.
TURN TO NEW PAGE
5. Let a and b be integers. Prove that if 𝑎 ≡ 7(𝑚𝑜𝑑8) and 𝑏 ≡ 3(𝑚𝑜𝑑8) then a. (15 pts) 𝑎 + 𝑏 ≡ 2(𝑚𝑜𝑑8) b. (15 pts.) 𝑎𝑏 ≡ 5(𝑚𝑜𝑑8) 6. (20 pts.) Let n be a natural numbers and let a, b, c and d be integers. Prove the following
proposition:
If )(mod nba and )(mod ndc , then ))(mod()( ndbca
TURN TO NEW PAGE 7. (20 pts.) Determine if the following statement is true or false. This means that if a
statement is true, then write a formal proof of that statement. However, if it is false then
provide a counter example that shows it is false
For all integers a, b, and c with 0a , if ba | and ca | then )(| cba .
TURN TO NEW PAGE 8. (10 pts) Use truth tables to prove one of DeMorgan’s Laws: that is prove
~(𝑃 ∧ 𝑄) ≡ ~𝑃 ∨ ~𝑄 TURN TO NEW PAGE Bonus: You will not lose points if you do this wrong. But you can earn extra points for correct
responses. Suppose each of the following statements is true:
I adopt abused malamutes, a big dog used for pulling sleds.
Pogo is Dr. Kim’s malamute puppy and Ilean is Dr Kim's 12 year old malamute.
Dr. Kim walked Pogo or Dr. Kim walked Ilean.
If Dr. Kim walked Ilean, then Pogo is not a puppy. If possible determine the truth value of the following statements. Carefully explain your
reasoning
a. Dr. Kim walked Pogo. b. Dr. Kim walked Ilean. c. Either Dr. Kim did not walk Pogo or Ilean.
Math 301 Summer 2020 Dr. Kimberly Vincent
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Directions for submission When you are done place your responses in order and create ONE pdf file.
Be sure the quality is good before uploading.
Be sure the pages are right side up. I will not grade any exams if the pages are upside down or sideways.
Name printed on first page in upper right hand corner.
Be sure file name contains your name and exam 1. For example my file name would be Dr.KimExam1.pdf.
Upload your ONE pdf file to blackboard to the dropbox for the exam under content prior to 7 pm Friday May 22, 2020