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Chapter 6 Direct Proof

Here are the mathematical definitions of even and odd.

  • An integer \(n \in \mathbb{Z}\) is even when there exists \(k \in \mathbb{Z}\) such that \(n = 2k\text{.}\)

  • An integer \(n \in \mathbb{Z}\) is odd when there exists \(k \in \mathbb{Z}\) such that \(n = 2k+1\text{.}\)

Use these definitions to prove the following statements.

Exercises Practice Problems

1. Odd to Even.

Prove that for all integers \(n \in \mathbb{Z}\text{,}\) if \(n\) is odd, then \(5n + 3 \) is even.

2. Any to Odd.

Prove that if \(n \in \mathbb{Z}\text{,}\) then \(5n^2+3n+7\) is odd.

3. Even/Odd Combo.

Prove that if \(a\) is any odd integer and \(b\) is any even integer, then \(2a+3b\) is an even integer.

4. Summing Squares.

Prove that if \(a\) is any odd integer and \(b\) is any even integer, then \(a^2+b^2\) is an odd integer.

5. Even and Odd.

Use a proof by contradiction to prove that no integer is both even and odd.