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The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. [2] In set-builder notation, ... Intersection distributes over union ...
The intersection of two sets and , denoted by , [3] is the set of all ... Union – Set of elements in any of some sets; References Further reading. Devlin ...
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.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Given any two sets A and B, a new set is formed by performing set operations between them. Their union A ∪ B is the set of all elements of A or B or both. The union of sets is defined by the logical operation of disjunction as = {() ()}, which is uses "or" in an inclusive sense: elements that are present in both sets is present in the union.
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
Using union and intersection: define [1] [2] = and = If these two sets are equal, then the set-theoretic limit of the sequence exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well.
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.