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:


 * $\displaystyle \sum_{u \mathop \in S \mathop \cup T} f \left({u}\right) = \sum_{s \mathop \in S} f(s) + \sum_{t \mathop \in T} f \left({t}\right) - \sum_{v \mathop \in S \mathop \cap T} f \left({v}\right)$

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