Unordered pair

In mathematics, an unordered pair or pair set is a set of the form {ab}, i.e. a set having two elements a and b with no particular relation between them, where {ab} = {ba}. In contrast, an ordered pair (ab) has a as its first element and b as its second element, which means (ab) ≠ (ba).

While the two elements of an ordered pair (ab) need not be distinct, modern authors only call {ab} 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 {aa} 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 {a1a2,... an}.[5][6][7]