Rule of sum

Statement

Let $A$ and $B$ be finite sets with $A \cap B = \varnothing$. Then $A \cup B$ is finite and $|A \cup B| = |A| + |B|$.

More generally, let $A_1, \dots, A_k$ be finite sets that are pairwise disjoint. Then $|A_1 \cup \dots \cup A_k| = |A_1| + \dots + |A_k|$.

Remarks

For the disjoint union $A \sqcup B$ of two arbitrary finite sets, the same formula $|A \sqcup B| = |A| + |B|$ holds. If $A$ and $B$ are not disjoint, the formula fails. The inclusion-exclusion formula for two sets gives the correct count.

Explanations

L1 human

If you choose one item from a first group or from a second group, and no item is in both groups, you add the sizes. For example, with 3 soups and 4 salads on a menu, there are $3 + 4 = 7$ ways to choose one starter.