Ordered pair - Definition and Overview

An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. An ordered pair with first element a and second element b is usually written as (a, b). The notation (a, b) is also used to denote an open interval on the real number line; context should make it clear which meaning is meant.

Two ordered pairs (a₁, b₁) and (a₂, b₂) are equal if and only if a₁ = a₂ and b₁ = b₂.

The set of all ordered pairs whose first element is in some set X and second element in some set Y is called the Cartesian product of X and Y, and written X × Y. Subsets of X × Y are binary relations.

Ordered triples and n-tuples (ordered lists of n terms) are defined recursively from this definition: an ordered triple (a,b,c) can be defined as (a , (b,c) ): two nested pairs. This approach is mirrored in programming languages: It is possible to represent a list of elements as a construction of nested ordered pairs. For example, the list (1 2 3 4 5) becomes (1, (2, (3, (4, (5, {}))))). The Lisp programming language uses such lists as its primary data structure.

In axiomatic set theory, where all mathematical objects are given set-theoretic definitions, the ordered pair (a, b) is defined as the set { {a}, {a, b} }. The statement that x is the first element of an ordered pair p can then be formulated as

∀ Y ∈ p : x ∈Y

and that x is the second element of p as

(∃ Y ∈ p : x ∈ Y) ∧ (∀ Y₁ ∈ p, ∀ Y₂ ∈ p : Y₁ ≠ Y₂ → (x ∉ Y₁ ∨ x ∉ Y₂)).

Note that this definition is still valid for the ordered pair p = (x,x) = { {x}, {x,x} } = { {x}, {x} } = { {x} }; in this case the statement (∀ Y₁ ∈ p, ∀ Y₂ ∈ p : Y₁ ≠ Y₂ → (x ∉ Y₁ ∨ x ∉ Y₂)) is trivially true, since it is never the case that Y₁ ≠ Y₂.

In the usual Zermelo-Fraenkel formulation of set theory including the axiom of regularity, ordered pairs (a, b) can also be defined as the set {a, {a, b}}. However, the axiom of regularity is required, since without it, it is possible to consider sets x and z such that x = {z}, z = {x}, and x ≠ z. Then we have that

(x, x) = {x, {x, x}} = {x,{x}} = {x, z} = {z, x} = {z, {z}} = {z, {z, z}} = (z, z)

although we want (x,x) ≠ (z,z).