Summation of Zero/Finite Set

Theorem

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

Let $S$ be a finite set.

Let $0 : S \to \mathbb A$ be the zero mapping.

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Then the summation of $0$ over $S$ equals zero:

$\ds \sum_{s \mathop \in S} 0 \left({s}\right) = 0$

Proof

At least three proofs are possible:

using the definition of summation and Indexed Summation of Zero

using Indexed Summation of Sum of Mappings

using Summation of Multiple of Mapping on Finite Set.

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