mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an unordered pair or pair set is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of the form , i.e. a set having two elements ''a'' and ''b'' with , where = . In contrast, an

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

(''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 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 is a

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the ''multiplicity'' of ...

. 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 In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

; its

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

(number of elements) is 2 or (if the two elements are not distinct) 1. In

axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, the existence of unordered pairs is required by an axiom, the

axiom of pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets ...

. More generally, an unordered ''n''-tuple is a set of the form .

Notes

References

* {{Citation , last1=Enderton , first1=Herbert , title=Elements of set theory , publisher=

Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...

, location=Boston, MA , isbn=978-0-12-238440-0 , year=1977. Basic concepts in set theory