Problem Set 6—Proof Methods: Contrapositive, Biconditionals, Contradiction

Purpose

This problem set develops your ability to use a number of proof techniques, specifically proof via the contrapositive, proofs of biconditionals, and proof by contradiction. It also reinforces habits of formal writing in proofs.

Background

This problem set is based on material in sections 3.2 and 3.3 of our textbook. We discussed, or will discuss, this material in class on October 3, 5, and 7.

Activity

Prove the following. All your proofs should be word-processed (i.e., not hand-written) and should follow the guidelines in Sundstrom’s text, particularly the new guideline of stating early in a proof what technique is being used.

Proposition 1

The proposition from exercise 3 in section 3.2 of our textbook, with the stipulation that the proposition is in fact true (for all integers a and b, if √(ab) ≠ (a+b)/2, then a ≠ b).

Proposition 2

For all integers n, 6 | n if and only if 2 | n and 3 | n.

Proposition 3

The proposition implied by exercise 8b in section 3.3 of our textbook. You will need to start by formulating the proposition, assuming that the special case in part a is true (The proposition from part a is “for all real numbers x, x + √2 is irrational, or -x + √2 is irrational”).

Proposition 4

Exercise 13b in section 13.3 of our textbook (for all real numbers a and b, if a ≠ 0 and b ≠ 0, then √(a²+b²) ≠ a+b).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.