Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An ntuple is a sequence (or ordered list) of n elements, where n is a nonnegative integer. There is only one 0tuple, an empty sequence, or empty tuple, as it is referred to. An ntuple is defined inductively using the construction of an ordered pair.
Mathematicians usually write 'tuples' by listing the elements within parentheses "" and separated by commas; for example, denotes a 5tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "⟨ ⟩". Braces "{ }" are only used in defining arrays in some programming languages such as C++ and Java, 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. In dynamically typed languages, such as Lisp, lists are commonly used as tuples.^{[citation needed]} 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]}
EtymologyEdit
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" (threefold) 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]}
Names for tuples of specific lengthsEdit
Table
 


PropertiesEdit
The general rule for the identity of two ntuples is
Thus a tuple has properties that distinguish it from a set.
 A tuple may contain multiple instances of the same element, so
tuple ; but set .  Tuple elements are ordered: tuple , but set .
 A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.
DefinitionsEdit
There are several definitions of tuples that give them the properties described in the previous section.
Tuples as functionsEdit
If we are dealing with sets, an ntuple can be regarded as a function, F, whose domain is the tuple's implicit set of element indices, X, and whose codomain, Y, is the tuple's set of elements. Formally:
where:
In slightly less formal notation this says:
Using this definition of tuples, it follows that there is only one tuple, the empty function.
Tuples as nested ordered pairsEdit
Another way of modeling tuples in Set Theory is as nested ordered pairs. This approach assumes that the notion of ordered pair has already been defined; thus a 2tuple
 The 0tuple (i.e. the empty tuple) is represented by the empty set .
 An ntuple, with n > 0, can be defined as an ordered pair of its first entry and an (n − 1)tuple (which contains the remaining entries when n > 1):
This definition can be applied recursively to the (n − 1)tuple:
Thus, for example:
A variant of this definition starts "peeling off" elements from the other end:
 The 0tuple is the empty set .
 For n > 0:
This definition can be applied recursively:
Thus, for example:
Tuples as nested setsEdit
Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory:
 The 0tuple (i.e. the empty tuple) is represented by the empty set ;
 Let be an ntuple , and let . Then, . (The right arrow, , could be read as "adjoined with".)
In this formulation:
ntuples of msetsEdit
In discrete mathematics, especially combinatorics and finite probability theory, ntuples arise in the context of various counting problems and are treated more informally as ordered lists of length n.^{[7]} ntuples whose entries come from a set of m elements are also called arrangements with repetition, permutations of a multiset and, in some nonEnglish literature, variations with repetition. The number of ntuples of an mset is m^{n}. 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 nfold Cartesian power S × S × ... S. Tuples are elements of this product set.
Type theoryEdit
In type theory, commonly used in programming languages, a tuple has a product type; this fixes not only the length, but also the underlying types of each component. Formally:
and the projections are term constructors:
The tuple with labeled elements used in the relational model has a record type. Both of these types can be defined as simple extensions of the simply typed lambda calculus.^{[9]}
The notion of a tuple in type theory and that in set theory are related in the following way: If we consider the natural model of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets (note: the use of italics here that distinguishes sets from types) such that:
and the interpretation of the basic terms is:
 .
The ntuple of type theory has the natural interpretation as an ntuple of set theory:^{[10]}
The unit type has as semantic interpretation the 0tuple.
See alsoEdit
NotesEdit
ReferencesEdit
 ^ "Algebraic data type  HaskellWiki". wiki.haskell.org.
 ^ "Destructuring assignment". MDN Web Docs.
 ^ "Does JavaScript Guarantee Object Property Order?". Stack Overflow.
 ^ "N‐tuple  Oxford Reference". oxfordreference.com. Retrieved 1 May 2015.
 ^
Blackburn, Simon (2016) [1994]. "ordered ntuple". The Oxford Dictionary of Philosophy. Oxford quick reference (3 ed.). Oxford: Oxford University Press. p. 342. ISBN 9780198735304. Retrieved 20170630.
ordered ntuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.
 ^ OED, s.v. "triple", "quadruple", "quintuple", "decuple"
 ^ D'Angelo & West 2000, p. 9
 ^ D'Angelo & West 2000, p. 101
 ^ Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. pp. 126–132. ISBN 0262162091.
 ^ Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint
SourcesEdit
 D'Angelo, John P.; West, Douglas B. (2000), Mathematical Thinking/ProblemSolving and Proofs (2nd ed.), PrenticeHall, ISBN 9780130144126
 Keith Devlin, The Joy of Sets. Springer Verlag, 2nd ed., 1993, ISBN 0387940944, pp. 7–8
 Abraham Adolf Fraenkel, Yehoshua BarHillel, Azriel Lévy, Foundations of school Set Theory, Elsevier Studies in Logic Vol. 67, 2nd Edition, revised, 1973, ISBN 0720422701, p. 33
 Gaisi Takeuti, W. M. Zaring, Introduction to Axiomatic Set Theory, Springer GTM 1, 1971, ISBN 9780387900247, p. 14
 George J. Tourlakis, Lecture Notes in Logic and Set Theory. Volume 2: Set Theory, Cambridge University Press, 2003, ISBN 9780521753746, pp. 182–193
External linksEdit
 The dictionary definition of tuple at Wiktionary