# 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

- Equivalence of Definitions of Ordered Pair
- Equality of Ordered Pairs
- Definition:Ordered Tuple as Ordered Set

- Results about
**ordered pairs**can be found**here**.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Relations - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory (Footnote $*$) - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{C}$ - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$. Sets - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Pairing - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**ordered pair** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom: Ordered Pairs - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**pair**