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Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
This diagram uses embedded text that can be easily translated using a text ... Inclusion/exclusion for three sets Based on Image: ... Inclusion–exclusion principle;
The inclusion–exclusion principle relates the size of the union of multiple sets, the size of each set, and the size of each possible intersection of the sets. The smallest example is when there are two sets: the number of elements in the union of A and B is equal to the sum of the number of elements in A and B , minus the number of elements ...
In the general case, for a word with n 1 letters X 1, n 2 letters X 2, ..., n r letters X r, it turns out (after a proper use of the inclusion-exclusion formula) that the answer has the form () , for a certain sequence of polynomials P n, where P n has degree n.
Inclusion–exclusion principle – Counting technique in combinatorics; Intersection (set theory) – Set of elements common to all of some sets; Iterated binary operation – Repeated application of an operation to a sequence; List of set identities and relations – Equalities for combinations of sets; Naive set theory – Informal set theories
This can be derived by using inclusion-exclusion to count the surjections from n to k and using the fact that the number of such surjections is ! {}. Additionally, this formula is a special case of the k th forward difference of the monomial x n {\displaystyle x^{n}} evaluated at x = 0:
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He noted the relation between such topics as inclusion-exclusion, classical number theoretic Möbius inversion, coloring problems and flows in networks. Since then, under the strong influence of Rota, the theory of Möbius inversion and related topics has become an active area of combinatorics.