Addition principle ($|A\bigsqcup B|=|A|+|B|$)

The rule of sum counts alternatives in disjoint finite sets by adding their sizes. For a disjoint union $A\bigsqcup B$, each element belongs to exactly one alternative, so $|A\bigsqcup B|=|A|+|B|$. Overlapping alternatives require accounting for repeated elements, as in the inclusion-exclusion principle. The groups indexed by first-return position in the Catalan recurrence are disjoint alternatives.