Problem Set 2 — Direct Proofs

Purpose

This problem set develops your ability to devise simple direct proofs and to write them according to standard conventions for formal proofs. The problem set thus addresses the following learning outcomes:

• 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 6, unravel abstract definitions, create intuition-forming examples or counterexamples, and prove conjectures
• 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 based on section 1.2 of our textbook. We discussed that material in classes on February 10 and 12.

Activity

Write formal proofs of the following conjectures. Each proof should be typed according to the usual mathematical conventions for formal proofs, including typeface rules (e.g., italic variable names, emphasized labels for theorems and proofs, etc.)

Conjecture 1

If \(n\) is an even integer, then \(3n + 2\) is an even integer.

Conjecture 2

(Exercise 11b from section 1.2 of our textbook. See the book for definitions and more details of the conjecture.)

If \(x_1\) and \(x_2\) are solutions to the quadratic equation \(ax^2 + bx + c = 0\), then \(x_1x_2 = \tfrac{c}{a}\).

Conjecture 3

(Exercise 10b from section 1.2 of our textbook. This question is based on definitions in Exercise 9, particularly definitions of “type 2” and “type 1” integers; you may want to do some of the things Exercise 9 describes in order to get comfortable with these definitions. See the book for more details.)

If \(a\) and \(b\) are type 2 integers, as defined in Exercise 9, then \(a + b\) is a type 1 integer, as defined in Exercise 9.

Problem 4

In the following parts, define a rational number to be any number that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \ne 0\).

Part A

Phineas Phoole claims that dividing a rational number by an integer produces a rational number. Phineas’s brother Phileas thinks about this claim for a minute, and then observes that there is one integer for which this claim doesn’t hold. What is that integer? Explain why the claim doesn’t hold for that integer, although your explanation doesn’t have to be a formal proof.

Part B

State formally as a theorem or conjecture a version of Phineas’s claim that takes into account Phileas’s exception, and give a formal proof of this modified claim.

(Although I often bring Phineas Phoole into questions that need someone to claim something silly, Parts A and B of this question actually do illustrate how a certain amount of math gets invented in the real world: someone thinks they have discovered a new theorem, upon thought (or attempted proof) it turns out to not quite hold, leading to a modified theorem that gets examined in turn, etc.)

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.