Definition:Signum Function

Natural Numbers
Let $$n \in \N$$.

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

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

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

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

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

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

Integers
Let $$n \in \Z$$.

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

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

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

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

Let $$\pi \in S_n$$.

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