Definition:Ordered Pair

Definition
The definition of a set does not take any account of the order in which the elements are listed.

That is, $$\left\{{a, b}\right\} = \left\{{b, a}\right\}$$, and the elements $$a$$ and $$b$$ have the same status - neither is distinguished above the other as being more "important".

An ordered pair is a two-element set together with an ordering.

In other words, one of the elements is distinguished above the other - it comes first.

Such a structure is written:
 * $$\left({a, b}\right)$$

and it means: "first $$a$$, then $$b$$".

Kuratowski formalisation
Kazimierz Kuratowski showed that the concept of a ordered pair can be formalised by the definition:


 * $$\left({a, b}\right) = \left\{{\left\{{a}\right\}, \left\{{a, b}\right\}}\right\}$$

Coordinates
In an ordered pair $$\left({a, b}\right)$$, the following terminology is used:
 * $$a$$ is called the first coordinate;
 * $$b$$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinates.

Alternative Notation
In the field of symbolic logic and modern treatments of set theory, the notation $$$$ is often seen to denote an ordered pair.

This notation is found in many textbooks and journal articles in set theory, including the widely referenced textbooks of Herbert B. Enderton and Patrick Suppes.

Some users even claim that $$$$ is the way to go, but such seem still to be in a minority.