Definition:Signum Function

Natural Numbers
Let $n \in \N$.

The signum function $\operatorname{sgn}: \N \to \N$ is defined as:
 * $\forall n \in \N: \operatorname{sgn} \left({n}\right) = \begin{cases}

0 & : n = 0 \\ 1 & : n > 0 \end{cases}$

It is clear from the definition of characteristic function that $\operatorname{sgn} \left({n}\right) = \chi_{\N^*} \left({n}\right)$, where $\N^* = \N - \left\{{0}\right\}$.

Signum Bar
The signum bar function $\overline {\operatorname{sgn}}: \N \to \N$ is defined as:
 * $\forall n \in \N: \overline {\operatorname{sgn}} \left({n}\right) = \begin{cases}

1 & : n = 0 \\ 0 & : n > 0 \end{cases}$

It is clear from the definition of characteristic function that $\overline {\operatorname{sgn}} \left({n}\right) = \chi_{\left\{{0}\right\}} \left({n}\right)$.

Integers
Let $n \in \Z$.

The signum function $\operatorname{sgn}: \Z \to \Z$ is defined as:
 * $\forall n \in \Z: \operatorname{sgn} \left({n}\right) = \begin{cases}

-1 & : n < 0 \\ 0 & : n = 0 \\ 1 & : n > 0 \end{cases}$

Thus $\operatorname{sgn}: \Z \to \Z$ is an extension of $\operatorname{sgn}: \N \to \N$.

Permutations
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\pi \in S_n$.

Then $\operatorname{sgn} \left({\pi}\right)$ is defined as the sign of $\pi$.