Union (set theory)

Short description: Set of elements in any of some sets

Union of two sets:
[math]\displaystyle{ ~A \cup B }[/math]

Union of three sets:
[math]\displaystyle{ ~A \cup B \cup C }[/math]

The union of A, B, C, D, and E is everything except the white area.

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 ([math]\displaystyle{ 0 }[/math]) 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.

## Contents

* 1 Union of two sets
* 2 Algebraic properties
* 3 Finite unions
* 4 Arbitrary unions
* 4.1 Notations
* 5 Notation encoding
* 6 See also
* 7 Notes
* 8 External links

## 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.[2] In set-builder notation,

[math]\displaystyle{ A \cup B = \\{ x: x \in A \text{ or } x \in B\\} }[/math].[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}
[math]\displaystyle{ A \cup B = \\{2,3,4,5,6, \dots\\} }[/math]

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.

## Algebraic properties

Binary union is an associative operation; that is, for any sets [math]\displaystyle{ A, B, \text{ and } C, }[/math] [math]\displaystyle{ A \cup (B \cup C) = (A \cup B) \cup C. }[/math]

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

Intersection distributes over union [math]\displaystyle{ A \cap (B \cup C) = (A \cap B)\cup(A \cap C) }[/math] and union distributes over intersection[2] [math]\displaystyle{ A \cup (B \cap C) = (A \cup B) \cap (A \cup C). }[/math]

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

## 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.

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]

## 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.[8] In symbols:

[math]\displaystyle{ x \in \bigcup \mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A. }[/math]

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.

### Notations

The notation for the general concept can vary considerably. For a finite union of sets [math]\displaystyle{ S_1, S_2, S_3, \dots , S_n }[/math] one often writes [math]\displaystyle{ S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n }[/math] or [math]\displaystyle{ \bigcup_{i=1}^n S_i }[/math]. Various common notations for arbitrary unions include [math]\displaystyle{ \bigcup \mathbf{M} }[/math], [math]\displaystyle{ \bigcup_{A\in\mathbf{M}} A }[/math], and [math]\displaystyle{ \bigcup_{i\in I} A_{i} }[/math]. The last of these notations refers to the union of the collection [math]\displaystyle{ \left\\{A_i : i \in I\right\\} }[/math], where I is an index set and [math]\displaystyle{ A_i }[/math] is a set for every [math]\displaystyle{ i \in I }[/math]. In the case that the index set I is the set of natural numbers, one uses the notation [math]\displaystyle{ \bigcup_{i=1}^{\infty} A_{i} }[/math], 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.

## Notation encoding

In Unicode, union is represented by the character U+222A ∪ UNION.[9] In TeX, [math]\displaystyle{ \cup }[/math] is rendered from `\cup` and [math]\displaystyle{ \bigcup }[/math] is rendered from `\bigcup`.

## See also

* Algebra of sets - Identities and relationships involving sets
* Alternation (formal language theory) − the union of sets of strings
* Axiom of union - Concept in axiomatic set theory
* Disjoint union - In mathematics, operation on sets
* Inclusion–exclusion principle - Counting technique in combinatorics
* Intersection (set theory) - Set of elements common to all of some sets
* Iterated binary operation - Repeated application of an operation to a sequence
* List of set identities and relations - Equalities for combinations of sets
* Naive set theory - Informal set theories
* Symmetric difference - Elements in exactly one of two sets

## Notes

1. ↑ Weisstein, Eric W. "Union". Wolfram Mathworld. http://mathworld.wolfram.com/Union.html.
2. ↑ 2.0 2.1 "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". https://www.probabilitycourse.com/chapter1/1_2_2_set_operations.php.
3. ↑ 3.0 3.1 Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01) (in en). Basic Set Theory. American Mathematical Soc.. ISBN 9780821827314. https://books.google.com/books?id=LBvpfEMhurwC.
4. ↑ deHaan, Lex; Koppelaars, Toon (2007-10-25) (in en). Applied Mathematics for Database Professionals. Apress. ISBN 9781430203483. https://books.google.com/books?id=2hM3-xxZC-8C&pg=PA24.
5. ↑ Halmos, P. R. (2013-11-27) (in en). Naive Set Theory. Springer Science & Business Media. ISBN 9781475716450. https://books.google.com/books?id=jV_aBwAAQBAJ.
6. ↑ Dasgupta, Abhijit (2013-12-11) (in en). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media. ISBN 9781461488545. https://books.google.com/books?id=u06-BAAAQBAJ.
7. ↑ "Finite Union of Finite Sets is Finite". https://proofwiki.org/wiki/Finite_Union_of_Finite_Sets_is_Finite.
8. ↑ 8.0 8.1 Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01) (in en). A Transition to Advanced Mathematics. Cengage Learning. ISBN 9781285463261. https://archive.org/details/transitiontoadva0000smit.
9. ↑ "The Unicode Standard, Version 15.0 - Mathematical Operators - Range: 2200–22FF". p. 3. https://www.unicode.org/charts/PDF/U2200.pdf.

## External links

* Hazewinkel, Michiel, ed. (2001), "Union of sets", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/u095390
* Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.

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