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

### Informal Definition

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:

$\tuple {a, b}$

and it means:

first $a$, then $b$.

### Kuratowski Formalization

The concept of an ordered pair can be formalized by the definition:

$\tuple {a, b} := \set {\set a, \set {a, b} }$

This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.

### Empty Set Formalization

The concept of an ordered pair can be formalized by the definition:

$\tuple {a, b} := \set {\set {\O, a}, \set {\set \O, b} }$

### Wiener Formalization

The concept of an ordered pair can be formalized by the definition:

$\tuple {a, b} := \set {\set {\O, \set a}, \set {\set b} }$

## Coordinates

Let $\tuple {a, b}$ be an ordered pair.

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 coordinate systems.

## Notation

In the field of symbolic logic and modern treatments of set theory, the notation $\sequence {a, b}$ is often seen to denote an ordered pair.

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

These notations are not used on $\mathsf{Pr} \infty \mathsf{fWiki}$, where $\tuple {a, b}$ is used exclusively.

## 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.

Some sources use the term ordered double.

## Also see

• Results about ordered pairs can be found here.