Proofs as Writing

Videos (~35 mins)

Integer Division and Modular Congruence (16:33)
Proofs as Writing (15:34)

Reading (~40 minutes)

Book of Proof, Chapter 5.

Focus on:

Congruence of integers (\(a \equiv b \mod{n}\)).
Proof style.

Warmup (~40 minutes)

Problem 1

Here are three theorems and attempted proofs. In each case, the proof has some problems: there might be an outright mistake, there might be a gap in the argument, or there might be a severe stylistic issue.

For each theorem please write a better proof.
After you’ve completed your first draft, please go through the list of twelve guidelines for mathematical writing supplied in the Book of Proof by Richard Hammack. Please check that your proofs are in accordance with these guidelines!
You’re done once each proof is mathematically correct and aligned with Hammack’s guidelines.

Hint: All of the supplied “bad” proofs contain useful ideas which can get you started on your own good proofs.

Theorem 1 (Part A) \(\mathbb{N} \subseteq \mathbb{Q}\).

Proof. \(x = x/1\).

Note: please use the definition of the set of rational numbers: \(\mathbb{Q} = \{ \frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0 \}\).

Theorem 2 (Part B) Let \(n \in \mathbb{Z}\) be odd. Then, \(n^2 \equiv 1 \mod{8}\).

Proof. We’ll use direct proof. Since \(n\) is odd, we can write \(n = 2k+1\) for some integer \(k\). Then, \(n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1\). It follows that \(8 \mid n^2 - 1\), and therefore \(n^2 \equiv 1 \mod{8}\).

Theorem 3 (Part C) For any integers \(a\), \(b\), \(c\), and \(p\), if \(a \equiv b \mod{p}\) and \(a \equiv c \mod{p}\) then \(b \equiv c \mod{p}\).

Proof. Direct proof. Since \(a\) is \(\equiv\) to \(b\) and \(a\) is also \(\equiv\) to \(c\), we can subtract the two equations. This will tell us that \(b\) is \(\equiv\) to \(c\).