# Union (set theory)

Short description: Set of elements in any of some sets

In set theory, the union (denoted by ∪) 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. A nullary union refers to a union of zero ($\displaystyle{ 0 }$) 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.

## 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 set-builder notation,

$\displaystyle{ A \cup B = \{ x: x \in A \text{ or } x \in B\} }$.

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then AB = {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}
$\displaystyle{ A \cup B = \{2,3,4,5,6, \dots\} }$

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, for any sets $\displaystyle{ A, B, \text{ and } C, }$ $\displaystyle{ A \cup (B \cup C) = (A \cup B) \cup C. }$

Thus, the parentheses may be omitted without ambiguity: either of the above can be written as $\displaystyle{ A \cup B \cup C. }$ Also, 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, $\displaystyle{ A \cup \varnothing = A, }$ for any set $\displaystyle{ A. }$ Also, the union operation is idempotent: $\displaystyle{ A \cup A = A. }$ All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union $\displaystyle{ A \cap (B \cup C) = (A \cap B)\cup(A \cap C) }$ and union distributes over intersection $\displaystyle{ A \cup (B \cap C) = (A \cup B) \cap (A \cup C). }$

The power set of a set $\displaystyle{ U, }$ 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 $\displaystyle{ A \cup B = \left(A^\text{c} \cap B^\text{c} \right)^\text{c}, }$ where the superscript $\displaystyle{ {}^\text{c} }$ denotes the complement in the universal set $\displaystyle{ U. }$

## 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:

$\displaystyle{ x \in \bigcup \mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A. }$

This idea subsumes the preceding sections—for example, ABC 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 $\displaystyle{ S_1, S_2, S_3, \dots , S_n }$ one often writes $\displaystyle{ S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n }$ or $\displaystyle{ \bigcup_{i=1}^n S_i }$. Various common notations for arbitrary unions include $\displaystyle{ \bigcup \mathbf{M} }$, $\displaystyle{ \bigcup_{A\in\mathbf{M}} A }$, and $\displaystyle{ \bigcup_{i\in I} A_{i} }$. The last of these notations refers to the union of the collection $\displaystyle{ \left\{A_i : i \in I\right\} }$, where I is an index set and $\displaystyle{ A_i }$ is a set for every $\displaystyle{ i \in I }$. In the case that the index set I is the set of natural numbers, one uses the notation $\displaystyle{ \bigcup_{i=1}^{\infty} A_{i} }$, which is 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.

## Notation encoding

In Unicode, union is represented by the character U+222A UNION. In TeX, $\displaystyle{ \cup }$ is rendered from \cup and $\displaystyle{ \bigcup }$ is rendered from \bigcup.