# Union (set theory)

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In set theory, the **union** (denoted by ∪) of a collection of sets is the set of all distinct elements in the collection.^{[1]} It is one of the fundamental operations through which sets can be combined and related to each other.

## Contents

## Union of two sets

The union of two sets *A* and *B* is the collection of points which are in *A* or in *B* or in both *A* and *B*. In symbols,

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}

If we are then to refer to a single element by the variable "*x*", then we can say that *x* is a member of the union if it is an element present in set *A* or in set *B*, or both.

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. 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.

## 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*.

These facts follow from analogous facts about logical disjunction.

## 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:

That this union of **M** is a set no matter how large a set **M** itself might be, is the content of the axiom of union in axiomatic set theory.

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.
The analogy between finite unions and logical disjunction extends to one between arbitrary unions and existential quantification.

### Notations

The notation for the general concept can vary considerably. For a finite union of sets one often writes . 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 series. When formatting is difficult, this can also be written "*A*_{1} ∪ *A*_{2} ∪ *A*_{3} ∪ ···".
(This last example, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.)

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

### Union and intersection

Intersection distributes over union, in the sense that

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.

Arbitrary union and intersection also satisfy the law

## See also

- Alternation (formal language theory), the union of sets of strings
- Cardinality
- Complement (set theory)
- Disjoint union
- Intersection (set theory)
- Iterated binary operation
- Naive set theory
- Symmetric difference

## Notes

## External links

- Weisstein, Eric W., "Union",
*MathWorld*. - {{#invoke:citation/CS1|citation

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- Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.