Finite Summation does not Change under Permutation
Jump to navigation
Jump to search
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.
$\blacksquare$