Union (set theory)

In set theory, the union of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
For explanation of the symbols used in this article, refer to the table of mathematical symbols.

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,
For example, if A = and B = then AB =. A more elaborate example is:
As another example, the number 9 is not contained in the union of the set of prime numbers and the set of even numbers, because 9 is neither prime nor even.
Sets cannot have duplicate elements, so the union of the sets and is. 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, for any sets A, B, and C,
The operations can be performed in any order, and the parentheses may be omitted without ambiguity. 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
and union distributes over intersection
Within a given universal set, union can be written in terms of the operations of intersection and complement as
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 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.

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 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. In symbols:
This idea subsumes the preceding sections—for example, ABC is the union of the collection. Also, if M is the empty collection, then the union of M is the empty set.


The notation for the general concept can vary considerably. For a finite union of sets one often writes or. Various common notations for arbitrary unions include ,, and, the last of which refers to the union of the collection where I is an index set and is a set for every. In the case that the index set I is the set of natural numbers, one uses a notation analogous to that of the infinite sums in series.
When the symbol "∪" is placed before other symbols instead of between them, it is usually rendered as a larger size.