Pairs

A pair is a 2-tuple consisting of two elements that are regarded as a single unit for the purposes of discussion or computation. In mathematics, a distinction is made between ordered pairs, in which the sequence matters, and unordered pairs, where the two elements are considered without regard to order.

An ordered pair (a,b) records that a is the first element and b is the second. In set theory, ordered pairs can be defined in terms of sets, for example Kuratowski’s definition (a,b) = {{a},{a,b}}; this encodes order via the structure of the set. The Cartesian product A×B is the set of all ordered pairs (a,b) with a in A and b in B. A fundamental property is that (a,b) = (c,d) if and only if a=c and b=d; in particular, (a,b) ≠ (b,a) in general when a ≠ b.

An unordered pair {a,b} contains exactly the two elements without specifying which is first. If a = b, the unordered pair collapses to the singleton {a}. Unordered pairs are common in elementary counting and in the study of sets where order is not important.

Beyond arithmetic and set theory, a pair often represents a point in a two-dimensional space, a two-component vector, or a simple data structure with two elements. In computer science, pairs (also called 2-tuples or simply pairs) are used to group related values, and many programming languages provide a built-in pair type or a generic product type.

In some contexts, pairs are used to encode or “pair up” elements, such as in pairing functions that map two inputs to a single output or in database keys and graph representations.