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. 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}.

Axiomatic definition in set/class theories

Constructor axiom:

$((A{\mbox{ set}})\land (B{\mbox{ set}}))\longrightarrow ((A\in \{A,B\})\land (B\in \{A,B\}))$

Destructor axiom:

$(C\in \{A,B\})\longrightarrow ((C=A)\lor (C=B))$

References

• Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.