Sum over Union of Finite Sets

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

Let $S$ and $T$ be finite sets.

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

Then we have the equality of summations over finite sets:


 * $\ds \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \mathop \in S} \map f s + \sum_{t \mathop \in T} \map f t - \sum_{v \mathop \in S \mathop \cap T} \map f v$

Proof
Follows from:


 * Mapping Defines Additive Function of Subalgebra of Power Set
 * Power Set is Algebra of Sets
 * Inclusion-Exclusion Principle

Also see

 * Inclusion-Exclusion Principle