The number of elements in the union of 3 sets is given by

where |X| denotes the cardinality of the set X. This may be written

where α‎₁ = |A| + |B| + |C|, α‎₂ = |A ∩ B| + |A ∩ C| + |B ∩ C| and α‎₃ = |A ∩ B ∩ C|. Now suppose that there are n sets instead of 3. Then the following, known as the inclusion–exclusion principle, holds:

where α‎_i is the sum of the cardinalities of the intersections of the sets taken i at a time.