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What is Principle of Inclusion and Exclusion mean?
In combinatorics, a branch of mathematics, 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
| A ∪ B | = | A | + | B | − | A ∩ B | , {\displaystyle |A\cup B|=|A|+|B|-|A\cap B|,}where A and B are two finite sets and |S| indicates the cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection.
The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets A, B and C is given by
| A ∪ B ∪ C | = | A | + | B | + | C | − | A ∩ B | − | A ∩ C | − | B ∩ C | + | A ∩ B ∩ C | . {\displaystyle |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|.}This formula can be verified by counting how many times each region in the Venn diagram figure is included in the right-hand side of the fo reference
Posted on 25 Dec 2024, this text provides information on Miscellaneous in Maths related to Maths. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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