Ordered pair

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In mathematics, an ordered pair is a collection (of objects) having two coordinates (or entries or projections), such that it is distinguishable, which object is the first coordinate (or first entry or left projection) of the pair and which object is the second coordinate (or second entry or right projection) of the pair. If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b.

Cartesian products and binary relations (and hence the ubiquitous functions) are defined in terms of ordered pairs.

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Generalities

Let (a1,b1) and (a2,b2) be two ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n terms). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. This approach is mirrored in computer programming languages that enable constructing a list of elements by nesting cons cells. For example, the list (1 2 3 4 5) becomes (1, (2, (3, (4, (5, {}))))). The Lisp programming language employs such lists as its primary data structure.

The set of all ordered pairs whose first element is in some set X and whose second element is in some set Y is called the Cartesian product of X and Y, and written X×Y. A binary relation over the field XY is a subset of X×Y.

If one wishes to employ the \ (a,b) notation for a different purpose (such as denoting open intervals on the real number line) the ordered pair may be denoted by the variant notation \left \langle a,b\right \rangle.

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