Problem Set 5 — Proofs Involving the Contrapositive or Biconditionals

Purpose

This problem set mainly lets you practice writing proofs that involve using the contrapositive, or proving a biconditional. Some of the proofs are chosen to use other material we’ve talked about recently. The problem set thus addresses the following learning outcomes:

• Outcome 2.1, Prove properties of and relationships between sets
• Outcome 5.1, Recognize claims amenable to proof using definitions and algebraic or other relationships pertinent to the subject of the claim, and construct the corresponding proofs
• Outcome 5.2, Recognize claims amenable to proof using the contrapositive and construct the corresponding proofs
• Outcome 7, Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics.

Background

This exercise is mainly based on section 3.2 of our textbook. We discussed that material in the “Proofs Via the Contrapositive” Canvas discussion and the March 10 class meeting.

Activity

Answer the following questions. Any formal proofs should be typed according to the usual mathematical conventions, including typeface rules (e.g., italic variable names, emphasized labels for theorems and proofs, etc.)

Question 1

Exercise 5 in section 3.2 of our textbook, giving a formal proof if you decide the statement is true. (Exercise 5 asks you to consider the statement that for all integers \(a\) and \(b\), if \(ab\) is even then \(a\) is even or \(b\) is even, and to either prove the statement true or give a counter-example to show that it is false.)

Question 2

Formally prove that for all integers \(n\), \(n\) is divisible by 6 if and only if \(n\) is divisible by 2 and \(n\) is divisible by 3. (Hint: the proposition explored in Question 1 might be helpful somewhere in your proof.)

Question 3

Give a formal proof for the conjecture that if \(A\) and \(B\) are sets and \(A \ne B\), then either \(A \nsubseteq B\) or \(B \nsubseteq A\).

Follow-Up

I will grade this exercise during one of your weekly individual meetings with me. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.