Editing Union (set theory)

{{short description|Set of elements in any of some sets}}
[[File:Venn0111.svg|thumb|200px|Union of two sets:<br /><math>~A \cup B</math>]]
[[File:Venn 0111 1111.svg |thumb|200px|Union of three sets:<br /><math>~A \cup B \cup C</math>]]
[[File:Example of a non pairwise disjoint family of sets.svg|200px|thumb|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 [[Set (mathematics)|sets]] is the set of all [[element (set theory)|element]]s in the collection.<ref>{{cite web |author=Weisstein |first=Eric W |title=Union |url=http://mathworld.wolfram.com/Union.html |url-status=live |archive-url=https://web.archive.org/web/20090207202412/http://mathworld.wolfram.com/Union.html |archive-date=2009-02-07 |access-date=2009-07-14 |publisher=Wolfram Mathworld}}</ref> It is one of the fundamental operations through which sets can be combined and related to each other. 
A '''{{visible anchor|nullary union|Nullary union}}''' refers to a union of [[Zero|zero ({{tmath|1= 0 }})]] sets and it is by definition equal to the [[empty set]]. 

For explanation of the symbols used in this article, refer to the [[List of mathematical symbols|table of mathematical symbols]].

== Binary union ==
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''.<ref name=":3">{{Cite web |title=Set Operations {{!}} Union {{!}} Intersection {{!}} Complement {{!}} Difference {{!}} Mutually Exclusive {{!}} Partitions {{!}} De Morgan's Law {{!}} Distributive Law {{!}} Cartesian Product |url=https://www.probabilitycourse.com/chapter1/1_2_2_set_operations.php |access-date=2020-09-05 |website=Probability Course}}</ref>  In [[set-builder notation]],
: <math>A  \cup B = \{ x: x \in A \text{  or  } x \in B\}</math>.<ref name=":0">{{Cite book|url=https://books.google.com/books?id=LBvpfEMhurwC|title=Basic Set Theory|last=Vereshchagin|first=Nikolai Konstantinovich|last2=Shen|first2=Alexander|date=2002-01-01|publisher=American Mathematical Soc.|isbn=9780821827314|language=en}}</ref>

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'' = {{mset|''x'' is an even [[integer]] larger than 1}}
: ''B'' = {{mset|''x'' is an odd integer larger than 1}}
: <math>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 number]]s {{mset|2, 3, 5, 7, 11, ...}} and the set of [[even number]]s {{mset|2, 4, 6, 8, 10, ...}}, because 9 is neither prime nor even.

Sets cannot have duplicate elements,<ref name=":0" /><ref>{{Cite book|url=https://books.google.com/books?id=2hM3-xxZC-8C&pg=PA24|title=Applied Mathematics for Database Professionals|last=deHaan|first=Lex|last2=Koppelaars|first2=Toon|date=2007-10-25|publisher=Apress|isbn=9781430203483|language=en}}</ref> so the union of the sets {{mset|1, 2, 3}} and {{mset|2, 3, 4}} is {{mset|1, 2, 3, 4}}. Multiple occurrences of identical elements have no effect on the [[cardinality]] of a set or its contents.

=== 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]].<ref>{{Cite book|url=https://books.google.com/books?id=u06-BAAAQBAJ|title=Set Theory: With an Introduction to Real Point Sets|last=Dasgupta|first=Abhijit|date=2013-12-11|publisher=Springer Science & Business Media|isbn=9781461488545|language=en}}</ref><ref>{{cite web |title=Finite Union of Finite Sets is Finite |url=https://proofwiki.org/wiki/Finite_Union_of_Finite_Sets_is_Finite |url-status=live |archive-url=https://web.archive.org/web/20140911224545/https://proofwiki.org/wiki/Finite_Union_of_Finite_Sets_is_Finite |archive-date=11 September 2014 |access-date=29 April 2018 |website=ProofWiki}}</ref>

== Notation ==
The notation for the general concept can vary considerably.  For a finite union of sets <math>S_1, S_2, S_3, \dots , S_n</math> one often writes <math>S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n</math> or <math display="inline">\bigcup_{i=1}^n S_i</math>.  Various common notations for arbitrary unions include <math display="inline">\bigcup \mathbf{M}</math>, <math display="inline">\bigcup_{A\in\mathbf{M}} A</math>, and <math display="inline">\bigcup_{i\in I} A_{i}</math>. The last of these notations refers to the union of the collection <math>\left\{A_i : i \in I\right\}</math>, where ''I'' is an [[index set]] and <math>A_i</math> is a set for every {{tmath|1= i \in I }}. In the case that the index set ''I'' is the set of [[natural number]]s, one uses the notation <math display="inline">\bigcup_{i=1}^{\infty} A_{i}</math>, which is analogous to that of the [[infinite sum]]s in series.<ref name=":1" />

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 {{unichar|222A|Union}}.<ref>{{cite web |title=The Unicode Standard, Version 15.0 – Mathematical Operators – Range: 2200–22FF |url=https://www.unicode.org/charts/PDF/U2200.pdf |website=[[Unicode]] |page=3}}</ref> In [[TeX]], <math>\cup</math> is rendered from <code>\cup</code> and <math display="inline">\bigcup</math> is rendered from <code>\bigcup</code>.

== Arbitrary union ==
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 (set theory)|class]] whose elements are sets, then ''x'' is an element of the union of '''M''' [[if and only if]] there is [[existential quantification|at least one]] element ''A'' of '''M''' such that ''x'' is an element of ''A''.<ref name=":1">{{Cite book |last=Smith |first=Douglas |url=https://archive.org/details/transitiontoadva0000smit |title=A Transition to Advanced Mathematics |last2=Eggen |first2=Maurice |last3=Andre |first3=Richard St |date=2014-08-01 |publisher=Cengage Learning |isbn=9781285463261 |language=en |url-access=registration}}</ref> In symbols:
: <math>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 {{mset|''A'', ''B'', ''C''}}. Also, if '''M''' is the empty collection, then the union of '''M''' is the empty set.

=== Formal derivation ===
In [[Zermelo–Fraenkel set theory]] (ZFC) and other set theories, the ability to take the arbitrary union of any sets is granted by the [[axiom of union]], which states that, given any set of sets <math>A</math>, there exists a set <math>B</math>, whose elements are exactly those of the elements of <math>A</math>. Sometimes this axiom is less specific, where there exists a <math>B</math> which contains the elements of the elements of <math>A</math>, but may be larger. For example if <math>A = \{ \{1\}, \{2\} \},</math> then it may be that  <math>B = \{ 1, 2, 3\}</math> since <math>B</math> contains 1 and 2. This can be fixed by using the [[axiom of specification]] to get the subset of <math>B</math> whose elements are exactly those of the elements of <math>A</math>. Then one can use the [[axiom of extensionality]] to show that this set is unique. For readability, define the binary [[Predicate (logic)|predicate]] <math>\operatorname{Union}(X,Y)</math> meaning "<math>X</math> is the union of <math>
Y</math>" or "<math>X = \bigcup Y</math>" as:

<math display="block">\operatorname{Union}(X,Y) \iff \forall x (x \in X \iff \exists y \in Y ( x \in y))</math>

Then, one can prove the statement "for all <math>Y</math>, there is a unique <math>X</math>, such that <math>X</math> is the union of <math>
Y</math>":

<math display="block">\forall Y \, \exists ! X (\operatorname{Union}(X,Y))</math>

Then, one can use an [[extension by definition]] to add the union operator <math>\bigcup A</math> to the [[Zermelo–Fraenkel set theory#Formal language|language of ZFC]] as:

<math display="block">\begin{align}
B = \bigcup A   & \iff \operatorname{Union}(B,A) \\
                & \iff \forall x (x \in B \iff \exists y \in Y(x \in y))
\end{align}</math>

or equivalently:

<math display="block">x \in \bigcup A \iff \exists y \in A \, (x \in y)</math>

After the union operator has been defined, the binary union <math>A \cup B</math> can be defined by showing there exists a unique set <math>C = \{A,B\}</math> using the [[axiom of pairing]], and defining <math>A \cup B = \bigcup \{A,B\}</math>. Then, finite unions can be defined inductively as:

<math display="block">\bigcup _ {i=1} ^ 0 A_i = \varnothing \text{, and } \bigcup_{i=1}^n A_i = \left(\bigcup_{i=1}^{n-1} A_i \right) \cup A_n</math>

== Algebraic properties ==
{{See also|List of set identities and relations|Algebra of sets}}

Binary union is an [[associative]] operation; that is, for any sets {{tmath|1= A, B, \text{ and } C }},
<math display="block">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 {{tmath|1= A \cup B \cup C }}. Also, union is [[commutative]], so the sets can be written in any order.<ref>{{Cite book |last=Halmos |first=P. R. |url=https://books.google.com/books?id=jV_aBwAAQBAJ |title=Naive Set Theory |date=2013-11-27 |publisher=Springer Science & Business Media |isbn=9781475716450 |language=en}}</ref>
The [[empty set]] is an [[identity element]] for the operation of union. That is, {{tmath|1= A \cup \varnothing = A }}, for any set {{tmath|1= A }}. Also, the union operation is idempotent: {{tmath|1= A \cup A = A }}.  All these properties follow from analogous facts about [[logical disjunction]].

Intersection distributes over union
<math display="block">A \cap (B \cup C) = (A \cap B)\cup(A \cap C)</math> 
and union distributes over intersection<ref name=":3" />
<math display="block">A \cup (B \cap C) = (A \cup B) \cap (A \cup C).</math>
The [[power set]] of a set {{tmath|1= U }}, together with the operations given by union, [[Intersection (set theory)|intersection]], and [[Complement (set theory)|complementation]], is a [[Boolean algebra (structure)|Boolean algebra]].  In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula
<math display="block">A \cup B = ( A^\complement \cap B^\complement )^\complement,</math>
where the superscript <math>{}^\complement</math> denotes the complement in the [[Universe (mathematics)|universal set]] {{tmath|1= U }}. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: <math>A \cap B = ( A^\complement \cup B^\complement )^\complement</math>. These two expressions together are called [[De Morgan's laws]].<ref>{{Cite web |title=MathCS.org - Real Analysis: Theorem 1.1.4: De Morgan's Laws |url=https://mathcs.org/analysis/reals/logic/proofs/demorgan.html |access-date=2024-10-22 |website=mathcs.org}}</ref><ref>{{Cite book |last=Doerr |first=Al |url=https://faculty.uml.edu/klevasseur/ads/s-laws-of-set-theory.html |title=ADS Laws of Set Theory |last2=Levasseur |first2=Ken |language=en-US}}</ref><ref>{{Cite web |title=The algebra of sets - Wikipedia, the free encyclopedia |url=https://www.umsl.edu/~siegelj/SetTheoryandTopology/The_algebra_of_sets.html |access-date=2024-10-22 |website=www.umsl.edu}}</ref>

== History and etymology ==
{{Further|History of set theory}}
The english word ''union'' comes from the term in [[middle French]] meaning "coming together", which comes from the [[Post-Classical Latin language|post-classical Latin]] ''unionem'', "oneness".<ref>{{Cite web |title=Etymology of "union" by etymonline |url=https://www.etymonline.com/word/union |access-date=2025-04-10 |website=etymonline |language=en-US}}</ref> The original term for union in set theory was ''Vereinigung'' (in german), which was introduced in 1895 by [[Georg Cantor]].<ref>{{Cite journal |last=Cantor |first=Georg |date=1895-11-01 |title=Beiträge zur Begründung der transfiniten Mengenlehre |url=https://link.springer.com/article/10.1007/BF02124929 |journal=Mathematische Annalen |language=de |volume=46 |issue=4 |pages=481–512 |doi=10.1007/BF02124929 |issn=1432-1807}}</ref> The english use of ''union'' of two sets in mathematics began to be used by at least 1912, used by [[James Pierpont (mathematician)|James Pierpont]].<ref>{{Cite book |last=Pierpont |first=James |url=https://archive.org/details/lecturesonthethe031634mbp/page/22/mode/2up?q=union |title=Lectures On The Theory Of Functions Of Real Variables Vol II |date=1912 |publisher=Ginn And Company |others=Osmania University, Digital Library Of India}}</ref><ref>''[[Oxford English Dictionary]]'', “union (''n.2''), sense III.17,” March 2025, https://doi.org/10.1093/OED/1665274057</ref> The symbol <math>\cup</math> used for union in mathematics was introduced by [[Giuseppe Peano]] in his ''[[Arithmetices principia, nova methodo exposita|Arithmetices principia]]'' in 1889, along with the notations for intersection <math>\cap</math>, set membership <math>\in</math>, and subsets <math>\subset</math>.<ref>{{Cite web |title=Earliest Uses of Symbols of Set Theory and Logic |url=https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/ |access-date=2025-04-10 |website=Maths History |language=en}}</ref>

== See also ==
{{Portal|Mathematics}}
* {{annotated link|Algebra of sets}}
* {{annotated link|Alternation (formal language theory)}} − the union of sets of strings
* {{annotated link|Axiom of union}}
* {{annotated link|Disjoint union}}
* {{annotated link|Inclusion–exclusion principle}}
* {{annotated link|Intersection (set theory)}}
* {{annotated link|Iterated binary operation}}
* {{annotated link|List of set identities and relations}}
* {{annotated link|Naive set theory}}
* {{annotated link|Symmetric difference}}

== Notes ==

{{Reflist|2}}

== External links ==
* {{springer|title=Union of sets|id=p/u095390}}
* [http://www.apronus.com/provenmath/sum.htm Infinite Union and Intersection at ProvenMath] De Morgan's laws formally proven from the axioms of set theory.

{{Set theory}}
{{Mathematical logic}}

[[Category:Basic concepts in set theory]]
[[Category:Boolean algebra]]
[[Category:Operations on sets]]
[[Category:Set theory]]