# 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.

## 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**.

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**.

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.

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.

## 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 - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**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