Union (set theory)

Union of two sets

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. In symbols,

A ∪ B = { x : x ∈ A  or  x ∈ B } .

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:

A = {x is an even integer larger than 1} B = {x is an odd integer larger than 1} A ∪ B = { 2 , 3 , 4 , 5 , 6 , … }

As another example, the number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.

Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.

Algebraic properties

Binary union is an associative operation; that is,

A ∪ (B ∪ C) = (A ∪ B) ∪ C.

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i.e., either of the above can be expressed equivalently as A ∪ B ∪ C). Similarly, union is commutative, so the sets can be written in any order.

The empty set is an identity element for the operation of union. That is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction.

Since sets with unions and intersections form a Boolean algebra, intersection distributes over union

A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )

and union distributes over intersection

A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) .

Within a given universal set, union can be written in terms of the operations of intersection and complement as

A ∪ B = ( A C ∩ B C ) C

where the superscript ^C denotes the complement with respect to the universal set.

Finite unions

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 A ∪ B ∪ C if and only if x is in at least one of A, B, and C.

In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the union set is a finite set.

Arbitrary unions

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set whose elements are themselves 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. In symbols:

x ∈ ⋃ M ⟺ ∃ A ∈ M ,   x ∈ A .

This idea subsumes the preceding sections, in that (for example) A ∪ B ∪ C is the union of the collection {A, B, C}. Also, if M is the empty collection, then the union of M is the empty set.

Notations

The notation for the general concept can vary considerably. For a finite union of sets S 1 , S 2 , S 3 , … , S n one often writes S 1 ∪ S 2 ∪ S 3 ∪ ⋯ ∪ S n or ⋃ i = 1 n S i . Various common notations for arbitrary unions include ⋃ M , ⋃ A ∈ M A , and ⋃ i ∈ I A i , the last of which refers to the union of the collection { A i : i ∈ I } where I is an index set and A i is a set for every i ∈ I . In the case that the index set I is the set of natural numbers, one uses a notation ⋃ i = 1 ∞ A i analogous to that of the infinite series.

Whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.