Definition:Sign of Ordered Tuple
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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.
Also denoted as
Some sources use $\map \sgn {x_1, x_2, \ldots, x_n}$.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Answers to Exercises: $46$