Editing Ordered pair (section)

{{short description|Pair of mathematical objects}}
[[File:Ellipse in coordinate system with semi-axes labelled.svg|thumb|300px|[[Analytic geometry]] associates to each point in the [[Euclidean plane]] an ordered pair. The red [[ellipse]] is associated with the set of all pairs (''x'',''y'') such that {{math|1={{sfrac|''x''<sup>2</sup>|4}} + ''y''<sup>2</sup> = 1}}.]]

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 ''[[unordered pair]]'', denoted {''a'', ''b''}, always equals the unordered pair {''b'', ''a''}.

Ordered pairs are also called [[tuple|2-tuples]], or [[sequence]]s (sometimes, lists in a computer science context) of length 2. Ordered pairs of [[scalar (mathematics)|scalars]] are sometimes called 2-dimensional [[Vector (mathematics and physics)|vectors]]. (Technically, this is an abuse of [[terminology]] since an ordered pair need not be an element of a [[vector space]].)
The entries of an ordered pair can be other ordered pairs, enabling the [[Recursive definition|recursive]] definition of ordered [[n-tuple|''n''-tuple]]s (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another.

In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ''second entry'' of the pair. Alternatively, the objects are called the first and second ''components'', the first and second ''coordinates'', or the left and right ''projections'' of the ordered pair.

[[Cartesian product]]s and [[binary relation]]s (and hence [[function (mathematics)|functions]]) are defined in terms of ordered pairs, cf. picture.