Ordered pair

Definition and properties

The first and second components of an ordered pair are distinguished; projection maps extract them: the first projection π1(a, b) = a and the second projection π2(a, b) = b. The pairing operation is injective in the sense that different pairs have different components. Ordered pairs generalize to n‑tuples, usually defined either as nested pairs or as functions from a finite index set to components.

Set‑theoretic encodings

In axiomatic set theory ordered pairs are commonly represented by sets so that pairs can be treated without adding new primitive types. A standard construction, due to Kuratowski, defines (a, b) as the set {{a}, {a, b}}. This encoding satisfies the required equality condition and works even when a = b. Other encodings have been proposed historically; the particular choice is a matter of convenience for foundational work and does not affect ordinary mathematics.

Uses and examples

Cartesian products are formed from ordered pairs: A × B = { (a, b) : a ∈ A, b ∈ B }. Relations are subsets of A × B; a function from A to B is a relation in which each a ∈ A occurs paired with exactly one b ∈ B. In coordinate geometry, points in the plane are ordered pairs of numbers (x, y) and vector spaces and product topologies are based on tuples. In computer science an ordered pair corresponds to a 2‑tuple or pair data structure, which may be immutable (as in many functional languages) or mutable (as in some array representations).

Remarks and generalizations

• Ordered pairs differ from unordered pairs: {a, b} = {b, a} in general, but (a, b) = (b, a) only when a = b.
• n‑tuples are iterated ordered pairs or maps from {1,...,n} to components; a 2‑tuple is an ordered pair.
• In category theory the product object generalizes ordered pairs: a product is determined up to unique isomorphism by universal properties rather than by a particular representation.

Notation varies: parentheses (a, b) and angle brackets <a, b> are both common. The simplicity and generality of ordered pairs make them one of the most widely used constructions across mathematics and its applications.