Unordered pair

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,... a_n}.^[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