Ordered pair

(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. To distinguish the two meanings, the ordered pair
is sometimes written as .)

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 usually defined as the Kuratowski pair (a, b)_K := {{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.

Notice that the set {1, 2} is the ordered pair (0, 1), since 1={0} and 2={0, 1}.

The above definition of an ordered pair is "adequate", in the sense that it satisfies the characteristic property that an ordered pair must have (namely: if (a,b)=(x,y), then a=x and b=y), but also arbitrary, as there are many other definitions which are no more complicated and would also be adequate. Examples for other possible definitions include