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. 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}.
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.
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 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 unorderedn tuple is a set of the form a1 a2 an NotesDuntsch Ivo Gediga Gunther 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 amp 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 VerlagReferencesEnderton Herbert 1977 Elements of set theory Boston MA Academic Press ISBN 978 0 12 238440 0
