Cardinality of Set Union

Theorem
Let $S_1, S_2, \ldots$ be sets.

Then:
 * $\left|{S_1 \cup S_2}\right| = \left|{S_1}\right| + \left|{S_2}\right| - \left|{S_1 \cap S_2}\right|$

Also:

and in general:

Proof
By Cardinality is Additive Function, we can apply the Inclusion-Exclusion Principle:

If $f: \mathcal S \to \R$ is an additive function, then: