2tuples

A 2-tuple, also called an ordered pair, is a pair of two elements arranged in a specific order. It is typically written as (a, b), where the first component is a and the second is b. The pair belongs to the Cartesian product A × B if a ∈ A and b ∈ B. Equality of 2-tuples is defined componentwise: (a, b) = (c, d) if and only if a = c and b = d. The order matters, so (a, b) and (b, a) are generally distinct.

Notation and interpretation vary by context. In many mathematical texts, angle brackets ⟨a, b⟩ are used as an alternative. In computer science and programming, 2-tuples are often treated as a single composite value or as a pair type, with common operations including projecting the first or second component.

Construction and variation: If A and B are the same set, the 2-tuple (a, b) lies in A × A. Finite examples include (3, 7) in ℕ × ℕ or ("x", 5) in {“x”} × ℕ. The collection of all such pairs from A and B forms the Cartesian product A × B, which in turn can be used to model various structured data.

Relation to other concepts: The set of all 2-tuples from X to Y is X × Y. A function from X to Y can be viewed as a special subset of X × Y that assigns to each x ∈ X exactly one y ∈ Y. In geometry, a 2-tuple represents a coordinate pair; in graph theory, an edge can be modeled as a 2-tuple of its endpoints; in programming, a 2-tuple is a basic data structure for pairing two values.

See also: n-tuple, Cartesian product, ordered pair, pair type, pairing function.