CSCI 301 L10 Worksheet

Lecture 10 - 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, \(7 = 4^2 - 3^2\).)