Definition:Sign of Ordered Tuple

Definition
Let $n \in \N$ be a natural number such that $n > 1$.

Let $\left({x_1, x_2, \ldots, x_n}\right)$ be an ordered $n$-tuple of real numbers.

Let $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ be the product of differences of $\left({x_1, x_2, \ldots, x_n}\right)$:
 * $\displaystyle \Delta_n \left({x_1, x_2, \ldots, x_n}\right) = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_i - x_j}\right)$

The sign of $\left({x_1, x_2, \ldots, x_n}\right)$ is defined and denoted as:


 * $\epsilon \left({x_1, x_2, \ldots, x_n}\right) := \operatorname{sgn} \left({\Delta_n}\right)$

where $\operatorname{sgn}$ denotes the signum function.

That is:
 * $\displaystyle \epsilon \left({x_1, x_2, \ldots, x_n}\right) := \operatorname{sgn} \left({\prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_i - x_j}\right)}\right)$

where:
 * $\operatorname{sgn} \left({\pi}\right) := \left[{x > 0}\right] - \left[{x < 0}\right]$
 * $\left[{x > 0}\right]$ etc. is Iverson's convention.

Also denoted as
Some sources use $\operatorname{sgn} \left({x_1, x_2, \ldots, x_n}\right)$.