Problem Set 6 — Proof by Contradiction

Purpose

This problem set gives you practice writing proofs by contradiction. It thus addresses the following learning outcomes:

• Outcome 5.3, Recognize claims amenable to proof by contradiction and construct the corresponding proofs. (Note: I think that proof by contradiction is a good method to use on this problem set, but if you use other proof methods that lead to valid proofs, I will replace or supplement this outcome with the ones relevant to methods you use.)
• 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 3.3 of our textbook. We discussed that material in class on March 15 and 17.

Activity

Prove the following propositions. Write the proofs as formal proofs, following the usual mathematical conventions, including typeface rules (e.g., italic variable names, emphasized labels for theorems and proofs, etc.)

Question 1

The generalized 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 \(x\), \(x + \sqrt{2}\) is irrational, or \(-x + \sqrt{2}\) is irrational”; see the textbook for more information).

Question 2

Exercise 16b in section 3.3 of Sundstrom’s text (show that for all integers \(a\) and \(b\), \(b^2 \ne 4a + 2\).)

Question 3

For all integers \(n\) and real numbers \(x\), \(n + x\) is irrational if and only if \(x\) is irrational.

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.