# Definition:Symmetric Mapping (Mapping Theory)

## Definition

Let $n \in \N$ be a natural number.

Let $S^n$ be an $n$-dimensional cartesian space on a set $S$.

Let $f: S^n \to T$ be a mapping from $S^n$ to a set $T$.

Then $f$ is a symmetric mapping if and only if:

$\map f {x_1, x_2, \dotsc, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \dotsc x_{\map \pi n} }$

for all permutations $\pi$ on $\set {1, 2, \dotsc n}$.

That is, a symmetric mapping is a mapping defined on a cartesian space whose values are preserved under permutation of its arguments.