union (set theory)#Arbitrary unions

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.{{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}} In set-builder notation,

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

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}}

: $A\; \backslash cup\; B\; =\; \backslash \{2,3,4,5,6,\; \backslash dots\backslash \}$

As another example, the number 9 is not contained in the union of the set of prime numbers {{mset|2, 3, 5, 7, 11, ...}} and the set of even numbers {{mset|2, 4, 6, 8, 10, ...}}, because 9 is neither prime nor even.

Sets cannot have duplicate elements,{{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}} 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.

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.{{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}}{{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}}

The notation for the general concept can vary considerably. For a finite union of sets $S\_1,\; S\_2,\; S\_3,\; \backslash dots\; ,\; S\_n$ one often writes $S\_1\; \backslash cup\; S\_2\; \backslash cup\; S\_3\; \backslash cup\; \backslash dots\; \backslash cup\; S\_n$ or $\backslash bigcup\_\{i=1\}^n\; S\_i$. Various common notations for arbitrary unions include $\backslash bigcup\; \backslash mathbf\{M\}$, $\backslash bigcup\_\{A\backslash in\backslash mathbf\{M\}\}\; A$, and $\backslash bigcup\_\{i\backslash in\; I\}\; A\_\{i\}$. The last of these notations refers to the union of the collection $\backslash left\backslash \{A\_i\; :\; i\; \backslash in\; I\backslash right\backslash \}$, where I is an index set and $A\_i$ is a set for every {{tmath|1= i \in I }}. In the case that the index set I is the set of natural numbers, one uses the notation $\backslash bigcup\_\{i=1\}^\{\backslash infty\}\; A\_\{i\}$, 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.

In Unicode, union is represented by the character {{unichar|222A|Union}}.{{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}} In TeX, $\backslash cup$ is rendered from \cup and $\backslash bigcup$ is rendered from \bigcup.

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.{{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}} In symbols:

: $x\; \backslash in\; \backslash bigcup\; \backslash mathbf\{M\}\; \backslash iff\; \backslash exists\; A\; \backslash in\; \backslash mathbf\{M\},\backslash \; x\; \backslash in\; A.$

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.

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 $A$, there exists a set $B$, whose elements are exactly those of the elements of $A$. Sometimes this axiom is less specific, where there exists a $B$ which contains the elements of the elements of $A$, but may be larger. For example if $A\; =\; \backslash \{\; \backslash \{1\backslash \},\; \backslash \{2\backslash \}\; \backslash \},$ then it may be that $B\; =\; \backslash \{\; 1,\; 2,\; 3\backslash \}$ since $B$ contains 1 and 2. This can be fixed by using the axiom of specification to get the subset of $B$ whose elements are exactly those of the elements of $A$. Then one can use the axiom of extensionality to show that this set is unique. For readability, define the binary predicate $\backslash operatorname\{Union\}(X,Y)$ meaning "$X$ is the union of $$

Y" or "$X\; =\; \backslash bigcup\; Y$" as:

$$\backslash operatorname\{Union\}(X,Y)\; \backslash iff\; \backslash forall\; x\; (x\; \backslash in\; X\; \backslash iff\; \backslash exists\; y\; \backslash in\; Y\; (\; x\; \backslash in\; y))$$

Then, one can prove the statement "for all $Y$, there is a unique $X$, such that $X$ is the union of $$

Y":

$$\backslash forall\; Y\; \backslash ,\; \backslash exists\; !\; X\; (\backslash operatorname\{Union\}(X,Y))$$

Then, one can use an extension by definition to add the union operator $\backslash bigcup\; A$ to the language of ZFC as:

$$\backslash begin\{align\}$$

B = \bigcup A & \iff \operatorname{Union}(B,A) \\

& \iff \forall x (x \in B \iff \exists y \in Y(x \in y))

\end{align}

or equivalently:

$$x\; \backslash in\; \backslash bigcup\; A\; \backslash iff\; \backslash exists\; y\; \backslash in\; A\; \backslash ,\; (x\; \backslash in\; y)$$

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

$$\backslash bigcup\; \_\; \{i=1\}\; ^\; 0\; A\_i\; =\; \backslash varnothing\; \backslash text\{,\; and\; \}\; \backslash bigcup\_\{i=1\}^n\; A\_i\; =\; \backslash left(\backslash bigcup\_\{i=1\}^\{n-1\}\; A\_i\; \backslash right)\; \backslash cup\; A\_n$$

{{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 }},

$$A\; \backslash cup\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cup\; C.$$

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.{{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}}

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

$$A\; \backslash cap\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cap\; B)\backslash cup(A\; \backslash cap\; C)$$

and union distributes over intersection

$$A\; \backslash cup\; (B\; \backslash cap\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cap\; (A\; \backslash cup\; C).$$

The power set of a set {{tmath|1= 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

$$A\; \backslash cup\; B\; =\; (\; A^\backslash complement\; \backslash cap\; B^\backslash complement\; )^\backslash complement,$$

where the superscript $\{\}^\backslash complement$ denotes the complement in the universal set {{tmath|1= U }}. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: $A\; \backslash cap\; B\; =\; (\; A^\backslash complement\; \backslash cup\; B^\backslash complement\; )^\backslash complement$. These two expressions together are called De Morgan's laws.{{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}}{{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}}{{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}}

{{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 unionem, "oneness".{{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}} The original term for union in set theory was Vereinigung (in german), which was introduced in 1895 by Georg Cantor.{{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}} The english use of union of two sets in mathematics began to be used by at least 1912, used by James Pierpont.{{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}}Oxford English Dictionary, “union (n.2), sense III.17,” March 2025, https://doi.org/10.1093/OED/1665274057 The symbol $\backslash cup$ used for union in mathematics was introduced by Giuseppe Peano in his Arithmetices principia in 1889, along with the notations for intersection $\backslash cap$, set membership $\backslash in$, and subsets $\backslash subset$.{{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}}

{{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}}

{{Reflist|2}}

• {{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.

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Category:Operations on sets

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