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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 ...
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:
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.
A more elaborate way of finding π(x) is due to Legendre (using the inclusion–exclusion principle): given x, if p 1, p 2,…, p n are distinct prime numbers, then the number of integers less than or equal to x which are divisible by no p i is
Inclusion (mineral), any material that is trapped inside a mineral during its formation; Inclusion bodies, aggregates of stainable substances in biological cells; Inclusion (cell), insoluble non-living substance suspended in a cell's cytoplasm; Inclusion (taxonomy), combining of biological species; Include directive, in computer programming
I added an example for the overall principle of inclusion / exclusion based on a deck of cards. Someone might want to add internal links. Antares5245 02:22, 15 May 2009 (UTC) The new section repeatedly had where appeared to be intended. Those are of course two different things.