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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".
Intersection distributes over union = () and union distributes over intersection [2] = (). The power set of a set U {\displaystyle U} , together with the operations given by union, intersection , and complementation , is a Boolean algebra .
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.
Intersection distributes over union and union distributes over intersection. That is, for any sets ,, and , one has = () = () Inside a universe , one may define the complement of to be the set of all elements of not in .
It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A). Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a, b), where a is a member of A and b is a member of B.
The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively. The properties below are stated without proof , but can be derived from a small number of properties taken as axioms .
Suppose that () = is a sequence of sets. The two equivalent definitions are as follows. 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.
The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets. In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure ...