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An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. An ordered pair with first element a and second element b is usually written as (a, b).
In pure set theory, where there are only sets, ordered pairs (a, b) can be simulated with the set { {a}, {a, b} }. The statement that x is the first element of an ordered pair p can then be formulated as
- for all Y in p : x in Y
and that x is the second element of p as
- (exist Y in p : x in Y) and (exist Y in p : x not in Y).
The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of said sets. Subsets thereof are mathematical relations.