Definitions become powerful when they are used inside proofs. Integer parity definitions and direct proofs show how “even” and “odd” are translated into algebraic forms. Once an integer is written as $2r$ or $2s+1$, addition and multiplication reveal the parity of the result. In this lesson, integer parity definitions and direct proofs are used to prove standard theorems about sums and products of even and odd integers.

Let $n$ be an integer. The integer $n$ is called even if there exists an integer $r$ such that $$\begin{equation*} n=2r. \end{equation*}$$ The integer $n$ is called odd if there exists an integer $s$ such that $$\begin{equation*} n=2s+1. \end{equation*}$$