Summation over Finite Set Equals Summation over Support

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 $\map \supp f$ be its support.

Then we have an equality of summations over finite sets:

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

Proof

Note that by Subset of Finite Set is Finite, $\map \supp f$ is indeed finite.

The result now follows from:

• Sum over Complement of Finite Set
• Sum of Zero over Finite Set
• Identity Element of Addition on Numbers

$\blacksquare$

Also see

• Definition:Summation over Set with Finite Support