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 zero () sets and it is by definition equal to the empty set.

- Union of two sets
- Algebraic properties
- Finite unions
- Arbitrary unions
- Notations
- Notation encoding
- See also
- Notes
- External links

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,

- .
^{ [3] }

For example, if *A* = {1, 3, 5, 7} and *B* = {1, 2, 4, 6, 7} 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}

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,^{ [3] }^{ [4] } 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.

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 *A* ∪ *B* ∪ *C*. 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: *A* ∪ *A* = *A*. All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union

and union distributes over intersection

^{ [2] }

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

where the superscript denotes the complement in the universal set *U*.

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*.

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:

This idea subsumes the preceding sections—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 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 .^{ [9] } The last of these notations 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 the notation , which is analogous to that of the infinite sums in series.^{ [8] }

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

In Unicode, union is represented by the character U+222A∪UNION. In TeX, is rendered from \cup.

- ↑ Weisstein, Eric W. "Union". Wolfram's Mathworld. Archived from the original on 2009-02-07. Retrieved 2009-07-14.
- 1 2 "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product".
*www.probabilitycourse.com*. Retrieved 2020-09-05. - 1 2 Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01).
*Basic Set Theory*. American Mathematical Soc. ISBN 9780821827314. - ↑ deHaan, Lex; Koppelaars, Toon (2007-10-25).
*Applied Mathematics for Database Professionals*. Apress. ISBN 9781430203483. - ↑ Halmos, P. R. (2013-11-27).
*Naive Set Theory*. Springer Science & Business Media. ISBN 9781475716450. - ↑ Dasgupta, Abhijit (2013-12-11).
*Set Theory: With an Introduction to Real Point Sets*. Springer Science & Business Media. ISBN 9781461488545. - ↑ "Finite Union of Finite Sets is Finite - ProofWiki".
*proofwiki.org*. Archived from the original on 11 September 2014. Retrieved 29 April 2018. - 1 2 Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01).
*A Transition to Advanced Mathematics*. Cengage Learning. ISBN 9781285463261. - ↑ "Comprehensive List of Set Theory Symbols".
*Math Vault*. 2020-04-11. Retrieved 2020-09-05.

Wikimedia Commons has media related to . Union (set theory) |

- "Union of sets",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.

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In mathematical logic and computer science, the **Kleene star** is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics it is more commonly known as the free monoid construction. The application of the Kleene star to a set *V* is written as *V*^{*}. It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions".

- If
*V*is a set of strings, then*V*^{*}is defined as the smallest superset of*V*that contains the empty string ε and is closed under the string concatenation operation. - If
*V*is a set of symbols or characters, then*V*^{*}is the set of all strings over symbols in*V*, including the empty string ε.

In mathematics, the **power set** of a set *S* is the set of all subsets of *S*, including the empty set and S itself. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(*S*), *P*(*S*), ℙ(*S*), ℘(*S*), or 2^{S}. The notation 2^{S} is used because given any set with exactly two elements, the powerset of *S* can be identified with the set of all functions from *S* into that set.

In mathematics, a **set** is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection Σ of subsets of *X* that includes *X* itself, is closed under complement, and is closed under countable unions.

In set theory, the **complement** of a set A, often denoted by *A*^{c}, are the elements not in A.

A **mathematical symbol** is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In mathematics, an **indicator function** or a **characteristic function** is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of X in A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

In mathematics, the **symmetric difference** of two sets, also known as the **disjunctive union**, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is .

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a **universe** is a collection that contains all the entities one wishes to consider in a given situation.

In topology and related branches of mathematics, the **Kuratowski closure axioms** are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.

In mathematics, a **partition of a set** is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.

In combinatorics, a branch of mathematics, the **inclusion–exclusion principle** is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as

In mathematics, **the algebra of sets**, not to be confused with the mathematical structure of *an* algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.

In mathematics, a **field of sets** is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an **algebra over ** that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.

In mathematics, a **Grothendieck universe** is a set *U* with the following properties:

- If
*x*is an element of*U*and if*y*is an element of*x*, then*y*is also an element of*U*. - If
*x*and*y*are both elements of*U*, then is an element of*U*. - If
*x*is an element of*U*, then*P*(*x*), the power set of*x*, is also an element of*U*. - If is a family of elements of
*U*, and if*I*is an element of*U*, then the union is an element of*U*.

In mathematics, an **iterated binary operation** is an extension of a binary operation on a set *S* to a function on finite sequences of elements of *S* through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names. In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted

In mathematics, the **intersection** of two sets *A* and *B*, denoted by *A* ∩ *B*, is the set containing all elements of *A* that also belong to *B*.

In mathematics, specifically set theory, the **Cartesian product** of two sets *A* and *B*, denoted *A* × *B*, is the set of all ordered pairs (*a*, *b*) where *a* is in *A* and *b* is in *B*. In terms of set-builder notation, that is

In computer science, more precisely in automata theory, a **rational set** of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra.

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