Signum Function is Primitive Recursive

Theorem

Let $\operatorname{sgn}: \N \to \N$ be defined as the signum function.

Then $\operatorname{sgn}$ is primitive recursive.

Signum Complement

Let $\overline {\operatorname{sgn}}: \N \to \N$ by defined as the signum-bar function.

Then $\overline {\operatorname{sgn}}$ is primitive recursive.

Proof

By Signum Function on Natural Numbers as Characteristic Function, $\map {\operatorname{sgn} } n = \chi_{\N^*}$, where $\N^* = \N \setminus \set 0$.

By Set of Non-Zero Natural Numbers is Primitive Recursive, $\N^*$ is primitive recursive.

Thus $\operatorname{sgn}$ is primitive recursive by definition of Primitive Recursive Set.

$\blacksquare$