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. In contrast, an ordered pair (a, b) has a as its first element and b as its second element.
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 2 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,... an}.
Famous quotes containing the word pair:
“Not the less does nature continue to fill the heart of youth with suggestions of his enthusiasm, and there are now men,—if indeed I can speak in the plural number,—more exactly, I will say, I have just been conversing with one man, to whom no weight of adverse experience will make it for a moment appear impossible, that thousands of human beings might exercise towards each other the grandest and simplest of sentiments, as well as a knot of friends, or a pair of lovers.”
—Ralph Waldo Emerson (1803–1882)