Finite Summation does not Change under Permutation

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Theorem

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $S$ be a finite set.

Let $f: S \to \mathbb A$ be a mapping.

Let $\sigma: S \to S$ be a permutation.


Then we have the equality of summations over finite sets:

$\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma s}$


Proof

This is a special case of Change of Variables in Summation over Finite Set.

$\blacksquare$