Inclusion–exclusion for two sets

Statement

Let $A$ and $B$ be finite sets. Then $$|A \cup B| + |A \cap B| = |A| + |B|.$$ Equivalently, $|A \cup B| = |A| + |B| - |A \cap B|$.

Remarks

If $A \cap B = \varnothing$, this is the rule of sum. For three sets: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$. The inclusion–exclusion principle is the version for $n$ sets.

Explanations

L1 human

When you add the sizes of $A$ and $B$, you count the elements of both sets twice. Subtract the size of $A \cap B$ once to correct this. For example, $12$ students play football, $8$ play chess, and $3$ do both, so $17$ students do at least one.