In
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 (''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 ...
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. T ...
; 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
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 referen ...
, location=Boston, MA , isbn=978-0-12-238440-0 , year=1977.
Basic concepts in set theory