Relation

Definition of Ordered Pair

What is an ordered pair ? What does it mean to arrange elements in some order ? Two things are involved in theis concept: One is that the elements must be distinguished from each other and one of them comes before the other. One way to capture these two characteristics is to note the elements that appear at or before each spot and to utilize them. Thus for the order a, b, the sets { a }, and { a, b } can be used to represent it. It is clearly diferent from the order b, a, which is represented by { b }, and { a, b }. That is in fact how an ordered pair is defined in naive set theory.

Definition: An ordered pair <a, b > is defined in terms of sets as follows: <a, b > = { { a }, { a, b } }.

Similarly an ordered n-tuple can be defined as
<a1, a2, ..., an > = { { a1 }, { a1, a2 }, ... , { a1, a2, ..., an } }.