Inclusion–Exclusion for Two Sets

When two sets overlap, simply adding their sizes counts the overlap twice. For two finite sets,

|A ∪ B| = |A| + |B| - |A ∩ B|.

The subtraction is not a correction trick to memorize. It follows from tracking how many times each region was counted: elements in only A or only B were counted once, while intersection elements were counted twice and therefore need one copy removed.

The Counting Explorer includes a small exact inclusion–exclusion workspace so you can change the region counts and see the union update immediately.