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
From the fact that Cardinality is an Additive Function, we can directly apply the Inclusion-Exclusion Principle:

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

$$ $$ $$ $$ $$