\begin{definition}[Definition:Sign of Permutation]
Let $n \in \N$ be a natural number.
Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.
Let $\pi$ be a permutation of $\N_n$.
Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$.
Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as:
:$\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$
The '''sign of $\pi \in S_n$''' is defined as:
:$\map \sgn \pi = \begin{cases}
\dfrac {\Delta_n} {\pi \cdot \Delta_n} & : \Delta_n \ne 0 \\
0 & : \Delta_n = 0 \end{cases}$
\end{definition}