twotuple

Twotuple, also called a 2-tuple or an ordered pair, is a mathematical object consisting of two components arranged in a specific order. It is the simplest nontrivial n-tuple; the first component is distinguished from the second, so (a,b) generally differs from (b,a). A twotuple can be viewed as an element of the Cartesian product A × B, where a ∈ A and b ∈ B.

Formally, an n-tuple of elements from a set X is a function f from the index set {1,…,n} to X. A 2-tuple is thus a function f: {1,2} → X with f(1)=a and f(2)=b. In set-theoretic practice, various encodings exist; for instance, the Kuratowski definition represents an ordered pair by nested sets. In everyday mathematics, (a,b) is taken as the primitive notation for the ordered pair.

Examples: (3,5) is a 2-tuple whose first component is 3 and second is 5; (x,y) is an arbitrary 2-tuple.

Relation to programming: many languages provide 2-tuple or pair types to group two values; these are used for returning two values from a function, storing coordinates, or as keys in data structures. Depending on the language, pairs may be mutable or immutable, and they often support projection operations such as first/second (fst/snd) or head/tail equivalents. The 2-tuple is the product of two types in type theory, and is a basic instance of the Cartesian product A × B; many algebraic constructions study properties of product types via projections.