\begin{definition}[Definition:Sign of Ordered Tuple]
Let $n \in \N$ be a natural number such that $n > 1$.
Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.
Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$:
:$\ds \map {\Delta_n} {x_1, x_2, \ldots, x_n} = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$
The '''sign of $\tuple {x_1, x_2, \ldots, x_n}$''' is defined and denoted as:
:$\map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\Delta_n}$
where $\sgn$ denotes the signum function.
That is:
:$\ds \map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j} }$
where:
:$\map \sgn \pi := \sqbrk {x > 0} - \sqbrk {x < 0}$
:$\sqbrk {x > 0}$ etc. is Iverson's convention.
\end{definition}