CSCI 301 L13 Worksheet

Lecture 13 - Exercises

Part A - Proof by Contradiction

1. Prove the following by contradiction: Proposition: Suppose \(n\) is an integer. If \(n\) is odd, then \(n^2\) is odd.

2. Consider the following proposition:

   Proposition: For every \(n \in \mathbb{Z}, 4 \not\mid (n^2 + 2)\).

   Set up the proof by contradiction. In other words, write the first line of the proof, beginning with “Suppose…”.

3. Complete the proof by contradiction of the proposition from #2.

4. Prove the following proposition: Proposition: Suppose \(a, b \in \mathbb{Z}\). If \(4 \mid (a^2 + b^2)\), then \(a\) and \(b\) are not both odd.

Book of Proof’s Style Guide

1. Begin each sentence with a word, not a mathematical symbol.
2. End each sentence with a period, even when the sentence ends with a mathematical symbol or expression.
3. Separate mathematical symbols and expressions with words.
4. Avoid misuse of symbols.
5. Avoid using unnecessary symbols.
6. Use the first person plural.
7. Use the active voice.
8. Explain each new symbol.
9. Watch out for “it”, unless it’s completely unambiguous what “it” refers to.
10. Since, because, as, for, so are used to mean that P is true (or assumed to be true) and as a consequence Q is true also.
11. Thus, hence, therefore, consequently precede a statement that follows logically from previous sentences or clauses.
12. Clarity is the gold standard of mathematical writing.

Definitions

Definition: An integer \(n\) is even if \(n = 2a\) for some integer \(a \in \mathbb{Z}\).

Observation: by (confusing) convention, definitions are usually stated as a conditional but meant to imply a biconditional. That is, we interpret the above definition to mean: \[ n \text{ is even} \Leftrightarrow \exists a \in Z, n = 2a \] What is the distinction? We are saying that \(n\) is not even if it does not satisfy the given property, where the unidirectional implication would be inconclusive about that case.

Some more definitions, just so we have something to play with in proofs.

Definition: An integer \(n\) is odd if \(n = 2a + 1\) for some integer \(a \in \mathbb{Z}\).

Definition: Suppose \(a\) and \(b\) are integers. We say that \(a\) divides \(b\), written \(a \mid b\), if \(b = ac\) for some \(c \in \mathbb{Z}\). In this case we also say that \(a\) is a divisor of \(b\), and that \(b\) is a multiple of \(a\).

Definition: Two integers have the same parity if they are both even or they are both odd. Otherwise they have opposite parity.

Definition/reminder: A real number \(x\) is rational if \(x = \frac{a}{b}\) for some integers \(a, b\). A number that is not rational is irrational.