Definition:Sign of Ordered Tuple

Definition

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.

Some sources use $\map \sgn {x_1, x_2, \ldots, x_n}$.