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 "$({\text{ }})$" and separated by commas; for example, $(2,7,4,1,7)$ 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 Java and Visual Basic, 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.^{[]} 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
The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., ntuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0tuple is called the null tuple. A 1tuple is called a singleton, a 2tuple is called an ordered pair and a 3tuple is a triple or triplet. n can be any nonnegative integer. For example, a complex number can be represented as a 2tuple, a quaternion can be represented as a 4tuple, an octonion can be represented as an 8tuple and a sedenion can be represented as a 16tuple.
Although these uses treat tuple as the suffix, the original suffix was ple as in "triple" (threefold) or "decuple" (tenfold). This originates from medieval Latin plus (meaning "more") related to Greek , which replaced the classical and late antique plex (meaning "folded"), as in "duplex".^{[6]}^{[a]}
Names for tuples of specific lengths
Table of names and varants for specific lengths 

Tuple length, $n$ 
Name 
Alternative names 

0 
empty tuple 
unit / empty sequence / null tuple 
1 
single 
singleton / monuple / monad 
2 
double 
dual / couple / (ordered) pair / twin / duad 
3 
triple 
treble / triplet / triad 
4 
quadruple 
quad / tetrad 
5 
quintuple 
pentuple / quint / pentad 
6 
sextuple 
hextuple 
7 
septuple 
heptuple 
8 
octuple 

9 
nonuple 

10 
decuple 

11 
undecuple 
hendecuple 
12 
duodecuple 

13 
tredecuple 

14 
quattuordecuple 

15 
quindecuple 

16 
sexdecuple 

17 
septendecuple 

18 
octodecuple 

19 
novemdecuple 

20 
vigintuple 

21 
unvigintuple 

22 
duovigintuple 

23 
trevigintuple 

24 
quattuorvigintuple 

25 
quinvigintuple 

26 
sexvigintuple 

27 
septenvigintuple 

28 
octovigintuple 

29 
novemvigintuple 

30 
trigintuple 

31 
untrigintuple 

40 
quadragintuple 

41 
unquadragintuple 

50 
quinquagintuple 

60 
sexagintuple 

70 
septuagintuple 

80 
octogintuple 

90 
nongentuple 

100 
centuple 

1,000 
milluple 


Properties
The general rule for the identity of two ntuples is
 $(a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})$ if and only if $a_{1}=b_{1},{\text{ }}a_{2}=b_{2},{\text{ }}\ldots ,{\text{ }}a_{n}=b_{n}.$
Thus a tuple has properties that distinguish it from a set.
 A tuple may contain multiple instances of the same element, so
tuple $(1,2,2,3)\neq (1,2,3)$; but set $\{1,2,2,3\}=\{1,2,3\}$.
 Tuple elements are ordered: tuple $(1,2,3)\neq (3,2,1)$, but set $\{1,2,3\}=\{3,2,1\}$.
 A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.
Definitions
There are several definitions of tuples that give them the properties described in the previous section.
Tuples as functions
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:
 $(a_{1},a_{2},\dots ,a_{n})\equiv (X,Y,F)$
where:
 ${\begin{aligned}X&=\{1,2,\dots ,n\}\\Y&=\{a_{1},a_{2},\ldots ,a_{n}\}\\F&=\{(1,a_{1}),(2,a_{2}),\ldots ,(n,a_{n})\}.\\\end{aligned}}$
In slightly less formal notation this says:
 $(a_{1},a_{2},\dots ,a_{n}):=(F(1),F(2),\dots ,F(n)).$
Tuples as nested ordered pairs
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 $\emptyset$.
 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):
 $(a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))$
This definition can be applied recursively to the (n  1)tuple:
 $(a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))$
Thus, for example:
 ${\begin{aligned}(1,2,3)&=(1,(2,(3,\emptyset )))\\(1,2,3,4)&=(1,(2,(3,(4,\emptyset ))))\\\end{aligned}}$
A variant of this definition starts "peeling off" elements from the other end:
 The 0tuple is the empty set $\emptyset$.
 For n > 0:
 $(a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n1}),a_{n})$
This definition can be applied recursively:
 $(a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})$
Thus, for example:
 ${\begin{aligned}(1,2,3)&=(((\emptyset ,1),2),3)\\(1,2,3,4)&=((((\emptyset ,1),2),3),4)\\\end{aligned}}$
Tuples as nested sets
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 $\emptyset$;
 Let $x$ be an ntuple $(a_{1},a_{2},\ldots ,a_{n})$, and let $x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)$. Then, $x\rightarrow b\equiv \{\{x\},\{x,b\}\}$. (The right arrow, $\rightarrow$, could be read as "adjoined with".)
In this formulation:
 ${\begin{array}{lclcl}&&&=&\emptyset \\&&&&\\(1)&=&\rightarrow 1&=&\{\{\},\{,1\}\}\\&&&=&\{\{\emptyset \},\{\emptyset ,1\}\}\\&&&&\\(1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\&&&=&\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\\&&&&\\(1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\&&&=&\{\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\},\\&&&&\{\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\},3\}\}\\\end{array}}$
ntuples of msets
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 theory
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:
 $(x_{1},x_{2},\ldots ,x_{n}):{\mathsf {T}}_{1}\times {\mathsf {T}}_{2}\times \ldots \times {\mathsf {T}}_{n}$
and the projections are term constructors:
 $\pi _{1}(x):{\mathsf {T}}_{1},~\pi _{2}(x):{\mathsf {T}}_{2},~\ldots ,~\pi _{n}(x):{\mathsf {T}}_{n}$
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 $S_{1},S_{2},\ldots ,S_{n}$ (note: the use of italics here that distinguishes sets from types) such that:
 $[\![{\mathsf {T}}_{1}]\!]=S_{1},~[\![{\mathsf {T}}_{2}]\!]=S_{2},~\ldots ,~[\![{\mathsf {T}}_{n}]\!]=S_{n}$
and the interpretation of the basic terms is:
 $[\![x_{1}]\!]\in [\![{\mathsf {T}}_{1}]\!],~[\![x_{2}]\!]\in [\![{\mathsf {T}}_{2}]\!],~\ldots ,~[\![x_{n}]\!]\in [\![{\mathsf {T}}_{n}]\!]$.
The ntuple of type theory has the natural interpretation as an ntuple of set theory:^{[10]}
 $[\![(x_{1},x_{2},\ldots ,x_{n})]\!]=(\,[\![x_{1}]\!],[\![x_{2}]\!],\ldots ,[\![x_{n}]\!]\,)$
The unit type has as semantic interpretation the 0tuple.
See also
Notes
 ^ Compare the etymology of ploidy, from the Greek for fold.
References
 ^ https://wiki.haskell.org/Algebraic_data_type
 ^ https://developer.mozilla.org/enUS/docs/Web/JavaScript/Reference/Operators/Destructuring_assignment
 ^ http://stackoverflow.com/questions/5525795/doesjavascriptguaranteeobjectpropertyorder
 ^ "Ntuple  Oxford Reference". oxfordreference.com. Retrieved 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 .
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. 126132. ISBN 0262162091.
 ^ Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint
Sources
 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. 78
 Abraham Adolf Fraenkel, Yehoshua BarHillel, Azriel Lévy, Foundations of set theory, Elsevier Studies in Logic Vol. 67, Edition 2, 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. 182193