Editing Union (set theory) (section)

== 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>