Inclusion-exclusion principle

In combinatorial mathematics, the inclusion exclusion principle (also known as the sieve principle) states that if A1, ..., An are finite sets, then where A denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting; in words: we can count the size of the union of sets A and B by adding A and B and then subtracting the size of their intersection. The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. When n 2 the exclusion of the pairwise intersections is (possibly) too severe, and the correct formula is as shown with alternating signs. This formula is attributed to Abraham de Moivre; it is sometimes also named for Daniel da Silva, Joseph Sylvester or Henri Poincaré. Inclusion exclusion illustrated for three sets Counts of each region with progressively more terms used for n = 4 For the case of three sets A, B, C the inclusion exclusion principle is illustrated in the graphic on the right.

Klappentext

In combinatorial mathematics, the inclusion-exclusion principle (also known as the sieve principle) states that if A1, ..., An are finite sets, then where |A| denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting; in words: we can count the size of the union of sets A and B by adding |A| and |B| and then subtracting the size of their intersection. The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. When n 2 the exclusion of the pairwise intersections is (possibly) too severe, and the correct formula is as shown with alternating signs. This formula is attributed to Abraham de Moivre; it is sometimes also named for Daniel da Silva, Joseph Sylvester or Henri Poincaré. Inclusion-exclusion illustrated for three sets Counts of each region with progressively more terms used for n = 4 For the case of three sets A, B, C the inclusion-exclusion principle is illustrated in the graphic on the right.