Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.[1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of sets and it is by definition equal to the empty set.

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

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 symbols,

Binary union is an associative operation; that is, for any sets A, B, and C,

Thus the parentheses may be omitted without ambiguity: either of the above can be written as ABC. Also, union is commutative, so the sets can be written in any order.[5] The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A. Also, the union operation is idempotent: AA = A. All these properties follow from analogous facts about logical disjunction.

The power set of a set U, together with the operations given by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula

One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of ABC if and only if x is in at least one of A, B, and C.

A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set.[6][7]

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A.[8] In symbols:

When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.