Exercise 1712Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Four-set inclusion-exclusion

#IE#four sets#union

Statement

For four sets, state the inclusion-exclusion formula.

Hint

Alternate sums of intersections of size $1,2,3,4$ .

Show solutionHide solution

Formula.
$|\cup A_i|=\sum|A_i|-\sum|A_i\cap A_j|+\sum|A_i\cap A_j\cap A_k|-\sum|A_1\cap A_2\cap A_3\cap A_4|.$

Interpretation. $+$ singles, $-$ pairs, $+$ triples, $-$ quadruple.

Conclusion. The pattern alternates with intersection size.