mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 no particular relation between them, where = . In contrast, an

ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...

(''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 Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance o ...

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 that e ...

. 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

(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 mathematics, ...

, 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 se ...

. 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 was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ...

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