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 $b$.

Coordinates
The elements of an ordered pair are called its coordinates.

Also known as
Some sources call this just a pair, taking the fact that it is ordered for granted.

However, this allows confusion with the concept of a doubleton set, so this usage is not recommended.

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.

In sources where the possibility of confusion is only minor, one can encounter $a \times b$ for $\left({a, b}\right)$ on an ad hoc basis.

Also see

 * Definition:Cartesian Product


 * Equality of Ordered Pairs


 * Definition:Ordered Tuple as Ordered Set