Parity: Every Natural Number with Zero Is Either Even or Odd, and Parity of Natural Numbers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let even and odd elements of $\mathbb{N}_{0}$ be as in Even and Odd Natural Numbers with Zero §even and Even and Odd Natural Numbers with Zero §odd.

¶ Every $n\in\mathbb{N}_{0}$ is even or odd, and not both.

¶ Let $n\in\mathbb{N}$. Then $n$ is even if and only if $n=2q$ for some $q\in\mathbb{N}$, and $n$ is odd if and only if $n+1=2q$ for some $q\in\mathbb{N}$.