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In set theory, the intersection of two sets and , denoted by , [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to . [2] Notation and terminology
However, the unit interval [0, 1] and the set of rational numbers Q are not almost disjoint, because their intersection is infinite. This definition extends to any collection of sets. A collection of sets is pairwise almost disjoint or mutually almost disjoint if any two distinct sets in the collection are almost disjoint. Often the prefix ...
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
The intersection of two signed sets is defined to be the signed set of elements that belong to both and have the same sign in both. According to this theorem, for any a collection of signed subsets of an n {\displaystyle n} -element set, all having cardinality r {\displaystyle r} and all pairs having a non-empty intersection, the number of ...
The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class.
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
Occasionally, the notation is used for the disjoint union of a family of sets, or the notation + for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.