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:


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

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