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In [[mathematics]], an '''unordered pair''' or '''pair set''' is a [[Set (mathematics)|set]] of the form {''a'', ''b''}, i.e. a set having two elements ''a'' and ''b'' with {{em|no particular relation between them}}, where {''a'', ''b''} = {''b'', ''a''}. In contrast, an [[ordered pair]] (''a'', ''b'') has ''a'' as its first element and ''b'' as its second element, which means (''a'', ''b'') ≠ (''b'', ''a''). |
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A '''binary set''' is a [[set]] with (exactly) two distinct elements, or, equivalently, a set whose [[cardinality]] is [[two]]. |
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While the two elements of an ordered pair (''a'', ''b'') need not be distinct, modern authors only call {''a'', ''b''} an unordered pair if ''a'' ≠ ''b''.<ref> |
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Examples: |
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{{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 }} |
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* The set {''a'',''b''} is binary. |
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</ref> |
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* The set {''a'',''a''} is not binary, since it is equivalent to the set {''a''}, which is a [[singleton (mathematics)|singleton]]. |
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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'', ''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. |
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A set with precisely two elements is also called a [[finite set|2-set]] or (rarely) a '''binary set'''. |
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In [[axiomatic set theory]], the existence of binary sets is a consequence of the [[axiom of empty set]] and the [[axiom of pairing]]. From these two axioms can be derived the existence of the singleton <nowiki>{{}}</nowiki>. From the axiom of empty set it is known that the set <nowiki>{}</nowiki> exists. From the axiom of pairing it is then known that the set <nowiki>{{},{{}}}</nowiki> exists, which contains both <nowiki>{}</nowiki> and <nowiki>{{}}</nowiki>. But the set <nowiki>{{},{{}}}</nowiki> is a binary set. |
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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) 1. |
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==See also== |
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* [[ordered pair]] |
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* [[binary relation]] |
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In [[axiomatic set theory]], the existence of unordered pairs is required by an axiom, the [[axiom of pairing]]. |
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More generally, an '''unordered '''''n'''''-tuple''' is a set of the form {''a''<sub>1</sub>, ''a''<sub>2</sub>,... ''a<sub>n</sub>''}.<ref> |
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{{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> |
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==Notes== |
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{{reflist}} |
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==References== |
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* {{Citation | last1=Enderton | first1=Herbert | title=Elements of set theory | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-238440-0 | year=1977}}. |
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Latest revision as of 03:40, 3 September 2024
In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a).
While the two elements of an ordered pair (a, b) need not be distinct, modern authors only call {a, b} an unordered pair if a ≠ b.[1][2][3][4] But for a few authors a singleton is also considered an unordered pair, although today, most would say that {a, 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.
A set with precisely two elements is also called a 2-set or (rarely) a binary set.
An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1.
In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.
More generally, an unordered n-tuple is a set of the form {a1, a2,... an}.[5][6][7]
Notes
[edit]- ^ Düntsch, Ivo; Gediga, Günther (2000), Sets, Relations, Functions, Primers Series, Methodos, ISBN 978-1-903280-00-3.
- ^ Fraenkel, Adolf (1928), Einleitung in die Mengenlehre, Berlin, New York: Springer-Verlag
- ^ Roitman, Judith (1990), Introduction to modern set theory, New York: John Wiley & Sons, ISBN 978-0-471-63519-2.
- ^ Schimmerling, Ernest (2008), Undergraduate set theory
- ^ Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3.
- ^ Rubin, Jean E. (1967), Set theory for the mathematician, Holden-Day
- ^ Takeuti, Gaisi; Zaring, Wilson M. (1971), Introduction to axiomatic set theory, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag
References
[edit]- Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.