In mathematics, an unordered pair or pair set is a set of the form {a,b}, i.e. a set having two elements a andb 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, 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.[citation needed] 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.[citation needed]

A set with precisely two elements is also called a 2-set or (rarely) a binary set.[citation needed]

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}.

Notes

  • Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN978-0-12-238440-0.