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Unordered pair

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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) unless a=b.

While the two elements of an ordered pair (a, b) need not be distinct, many authors require the elements of an unordered pair {a, b} to be distinct.[1][2][3][4] However, unordered pairs whose two elements are not required to be distinct are also sometimes used, especially in the context of graph theory, where they can represent loops in undirected graphs.[5]

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}.[6][7][8]

Notes#edit

  1. ^ Düntsch, Ivo; Gediga, Günther (2000), Sets, Relations, Functions, Primers Series, Methodos, ISBN 978-1-903280-00-3.
  2. ^ Fraenkel, Adolf (1928), Einleitung in die Mengenlehre, Berlin, New York: Springer-Verlag
  3. ^ Roitman, Judith (1990), Introduction to modern set theory, New York: John Wiley & Sons, ISBN 978-0-471-63519-2.
  4. ^ Schimmerling, Ernest (2008), Undergraduate set theory
  5. ^ Bondy, J. A.; Murty, U. S. R. (1976), Graph Theory With Applications, North-Holland, p. 1
  6. ^ Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3.
  7. ^ Rubin, Jean E. (1967), Set theory for the mathematician, Holden-Day
  8. ^ 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.


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