Summation of Zero/Set
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 set.
Let $0: S \to \mathbb A$ be the zero mapping.
Then the summation with finite support of $0$ over $S$ equals zero:
- $\ds \sum_{s \mathop \in S} \map 0 s = 0$
Proof
By Support of Zero Mapping, the support of $0$ is empty.
By Empty Set is Finite, the support of $0$ is indeed finite.
- $\ds \sum_{s \mathop \in S} \map 0 s = \sum_{s \mathop \in \O} \map 0 s = 0$
$\blacksquare$