CSCI 301 L07 Worksheet

Lecture 7 - Exercises

Part A - Direct Proof

Use direct proof to prove each of the following statements.

1. Suppose \(x, y \in \mathbb{Z}\). If \(x\) is even, then \(xy\) is even.
2. Suppose \(a\), \(b\) and \(c\) are integers. If \(a|b\) and \(a|c\), then \(a|(b+c)\).

Part B - Cases

Use direct proof to prove each of the following statments.

1. If two integers have the same parity, then their sum is even.
2. If two integers have opposite parity, then their sum is odd.

Part C - More

1. Let \(x\) and \(y\) be positive numbers. If \(x \le y\), then \(\sqrt{x} \le \sqrt{y}\).
2. Any odd number is the difference of two squares. (For example, \(7 = 4^2 - 3^2\).)

Definitions

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

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 \(a | b\), 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.