The Inclusion-Exclusion Principle — Two Sets

Section 3 already hinted at a problem: simply adding $n(A) + n(B)$ over-counts any elements shared between $A$ and $B$, because those elements get counted once inside $n(A)$ and again inside $n(B)$. The inclusion-exclusion principle for two sets fixes this by subtracting the overlap back out exactly once:

If $A$ and $B$ happen to be disjoint (Section 5), $n(A \cap B) = 0$ and the formula collapses to the simple $n(A \cup B) = n(A) + n(B)$ from before — disjoint sets are exactly the special case where nothing needs subtracting. …