# Signum Complement Function on Natural Numbers as Characteristic Function

## Definition

Let $\overline {\operatorname{sgn}}: \N \to \N$ be the signum complement function on the natural numbers.

Let $\chi_{\left\{{0}\right\}}: \N \to \left\{{0, 1}\right\}$ be the characteristic function of $\left\{{0}\right\}$.

Let $n \in \N$.

Then:

$\overline {\operatorname{sgn}} \left({n}\right) = \chi_{\left\{{0}\right\}} \left({n}\right)$

## Proof

The signum complement function $\overline {\operatorname{sgn}}: \N \to \N$ is defined as:

$\forall n \in \N: \operatorname{sgn} \left({n}\right) = \begin{cases} 1 & : n = 0 \\ 0 & : n > 0 \end{cases}$

The result follows by definition of the characteristic function.

$\blacksquare$