The unordered pair of and , denoted , has the property that if and only if and or and . In other words, the unordered pair is the same as the ordered pair , except that presentation order does not matter.
A more transparent terminology calls an unordered pair a pair set, which allows a pair to unambiguously be an ordered pair (as is usual in current usage), however the ordered/unordered distinction is well entrenched in the literature.
Unordered pairs are commonly defined as subsets, as follows:
If is a set and and are elements of , then the unordered pair or pair set is the subset of with the property that if and only if or . Note that , a singleton.
In material set theory, we may apply this when and are not previously given as elements of any set . In that case, the existence of the unordered pair is given by the axiom of pairing.
The set of all unordered pairs of elements of may be denoted . Classically (using excluded middle), is the internal disjoint union ; in other words, every unordered pair is either a -element set (a singleton) or a -element set.
Relation to ordered pairs
The unordered pair should not be confused with the ordered pair . In particular, , while (if ). In material set theory, has a direct definition, but must be coded in a complicated way (traditionally as ). On the other hand, ordered pairs are more natural in structural set theory.
However, the two are somewhat related:
- As just seen, the usual encoding of an ordered pair of pure sets as a pure set (due to Kuratowski) involves unordered pairs. Conversely, the set of unordered pairs of elements of may be constructed as a quotient set of the set of ordered pairs of elements of (by the equivalence relation generated by declaring that ).
- Just as unordered pairs (through the axiom of pairing) are needed to get anywhere in material set theory, so some axiom related to ordered pairs is needed to get anywhere in structural set theory. (In ETCS, this is the axiom of the (Cartesian) product; in SEAR, this is the axiom of tabulations.)
The term ‘pairing’ in the Lab usually refers to ordered pairs.