Tuple
In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively.
For the musical term, see Tuplet. "Octuple" redirects here. For the boat, see Octuple scull. "Duodecuple" redirects here. For the musical technique, see Twelve-tone technique. "Sextuple" redirects here. For the sporting achievement of association football, see Sextuple (association football).
Tuple may be formally defined from ordered pairs by recurrence by starting from ordered pairs; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element.
Tuples are usually written by listing the elements within parentheses "( )", separated by a comma and a space; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "⟨ ⟩". Braces "{ }" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term tuple can often occur when discussing other mathematical objects, such as vectors.
In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types,[1] tightly associated with algebraic data types, pattern matching, and destructuring assignment.[2] Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label.[3] A few programming languages combine ordered tuple product types and unordered record types into a single construct, as in C structs and Haskell records. Relational databases may formally identify their rows (records) as tuples.
Tuples also occur in relational algebra; when programming the semantic web with the Resource Description Framework (RDF); in linguistics;[4] and in philosophy.[5]
Etymology[edit]
The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0-tuple is called the null tuple or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple.
Although these uses treat ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".[6][a]
n-tuples of m-sets[edit]
In discrete mathematics, especially combinatorics and finite probability theory, n-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length n.[7] n-tuples whose entries come from a set of m elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of n-tuples of an m-set is mn. This follows from the combinatorial rule of product.[8] If S is a finite set of cardinality m, this number is the cardinality of the n-fold Cartesian power S × S × ⋯ × S. Tuples are elements of this product set.
