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<p><b>New page</b></p><div>In [[mathematics]], an '''unordered pair''' or '''pair set''' is a [[Set (mathematics)|set]] of the form {''a'',&nbsp;''b''}, i.e. a set having two elements ''a'' and&nbsp;''b'' with {{em|no particular relation between them}}, where {''a'',&nbsp;''b''} = {''b'',&nbsp;''a''}. In contrast, an [[ordered pair]] (''a'',&nbsp;''b'') has ''a'' as its first element and ''b'' as its second element, which means (''a'',&nbsp;''b'') ≠ (''b'',&nbsp;''a''). <br />
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While the two elements of an ordered pair (''a'',&nbsp;''b'') need not be distinct, modern authors only call {''a'',&nbsp;''b''} an unordered pair if ''a''&nbsp;≠&nbsp;''b''.<ref><br />
{{Citation | last1=Düntsch | first1=Ivo | last2=Gediga | first2=Günther | title=Sets, Relations, Functions | publisher=Methodos | series=Primers Series | isbn=978-1-903280-00-3 | year=2000}}.</ref><ref>{{Citation | last1=Fraenkel | first1=Adolf | title=Einleitung in die Mengenlehre | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1928}}</ref><ref>{{Citation | last1=Roitman | first1=Judith | title=Introduction to modern set theory | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-63519-2 | year=1990 | url-access=registration | url=https://archive.org/details/introductiontomo0000roit }}.</ref><ref>{{Citation | last1=Schimmerling | first1=Ernest | title=Undergraduate set theory | year=2008 }}<br />
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But for a few authors a [[Singleton (mathematics)|singleton]] is also considered an unordered pair, although today, most would say that {''a'',&nbsp;''a''} is a [[multiset]]. It is typical to use the term unordered pair even in the situation where the elements a and b could be equal, as long as this equality has not yet been established.<br />
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A set with precisely two elements is also called a [[finite set|2-set]] or (rarely) a '''binary set'''.<br />
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An unordered pair is a [[finite set]]; its [[cardinality]] (number of elements) is 2 or (if the two elements are not distinct)&nbsp;1.<br />
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In [[axiomatic set theory]], the existence of unordered pairs is required by an axiom, the [[axiom of pairing]].<br />
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More generally, an '''unordered '''''n'''''-tuple''' is a set of the form {''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,...&nbsp;''a<sub>n</sub>''}.<ref><br />
{{Citation | last1=Hrbacek | first1=Karel | last2=Jech | first2=Thomas | author2-link=Thomas Jech | title=Introduction to set theory | publisher=Dekker | location=New York | edition=3rd | isbn=978-0-8247-7915-3 | year=1999}}.</ref><ref>{{Citation | last1=Rubin | first1=Jean E. |author1-link=Jean E. Rubin | title=Set theory for the mathematician | publisher=Holden-Day | year=1967}}</ref><ref>{{Citation | last1=Takeuti | first1=Gaisi | last2=Zaring | first2=Wilson M. | title=Introduction to axiomatic set theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | year=1971}}</ref><br />
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==Notes==<br />
{{reflist}}<br />
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==References==<br />
* {{Citation | last1=Enderton | first1=Herbert | title=Elements of set theory | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-238440-0 | year=1977}}.<br />
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[[Category:Basic concepts in set theory]]</div>imported>Fgnievinski