Unordered pair: Difference between revisions
Georgydunaev (talk | contribs) added axiomatic definition which doesn't specify explicitly what {A,B} for proper classes is, so these axioms are universal for different systems Tag: Reverted |
Undid revision 1225961304 by Georgydunaev (talk) Lacks context and references. |
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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> |
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> |
{{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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== Axiomatic definition in set/class theories == |
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Constructor axiom: |
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:<math>((A\mbox{ set})\land (B\mbox{ set}))\longrightarrow ((A\in\{A,B\})\land (B\in\{A,B\}))</math> |
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Destructor axiom: |
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:<math>(C\in\{A,B\})\longrightarrow ((C=A)\lor(C=B))</math> |
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==Notes== |
==Notes== |
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Revision as of 05:30, 1 June 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
- ^ 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
- Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.