Definition:Ordered Pair

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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 refer to an ordered pair as 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 couple, but that has a completely different meaning in the context of mechanics.

Some sources use the term ordered double.


Also see

  • Results about ordered pairs can be found here.


Sources

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