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Tuple
A tuple is a finite ordered list of elements. In mathematics, an ntuple is a sequence (or ordered list) of <math>n</math> elements, where <math>n</math> is a nonnegative integer. There is only one 0tuple, an empty sequence. An <math>n</math>tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "<math>(\text{ })</math>" and separated by commas; for example, <math>(2, 7, 4, 1, 7)</math> denotes a 5tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "<math>\langle\text{ }\rangle</math>". Braces "{ }" are never used for tuples, as they are the standard notation for sets. Tuples are often used to describe other mathematical objects, such as vectors. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra. Tuples are also used in relation to programming the semantic web with Resource Description Framework or RDF. Tuples are also used in linguistics^{[1]} and philosophy.^{[2]}
Contents
Etymology
The term originated as an abstraction of the sequence: single, 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. A 1‑tuple is called a singleton, a 2‑tuple is called an ordered pair and a 3‑tuple is a triple or triplet. n can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple, 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 ‑tuple as the suffix, the original suffix was ‑ple as in "triple" (threefold) or "decuple" (ten‑fold). This originates from a medieval Latin suffix ‑plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".^{[3]}
Names for tuples of specific lengths
Tuple Length <math>n</math>  Name  Alternative names 

0  empty tuple  unit 
1  single  singleton 
2  double  couple / pair / dual / twin / product 
3  triple  treble / triplet 
4  quadruple  quad 
5  quintuple  
6  sextuple  
7  septuple  
8  octuple  
9  nonuple  
10  decuple  
11  undecuple  hendecuple 
12  duodecuple  
13  tredecuple  
100  centuple 
Properties
The general rule for the identity of two <math>n</math>tuples is
 <math>(a_1, a_2, \ldots, a_n) = (b_1, b_2, \ldots, b_n)</math> if and only if <math>a_1=b_1,\text{ }a_2=b_2,\text{ }\ldots,\text{ }a_n=b_n.</math>
Thus a tuple has properties that distinguish it from a set.
 A tuple may contain multiple instances of the same element, so
tuple <math>(1,2,2,3) \neq (1,2,3)</math>; but set <math>\{1,2,2,3\} = \{1,2,3\}</math>.  Tuple elements are ordered: tuple <math>(1,2,3) \neq (3,2,1)</math>, but set <math>\{1,2,3\} = \{3,2,1\}</math>.
 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 <math>n</math>tuple 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:
 <math>(a_1, a_2, \dots, a_n) \equiv (X,Y,F)</math>
where:
 <math>
\begin{align} 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{align} </math>
In slightly less formal notation this says:
 <math> (a_1, a_2, \dots, a_n) := (F(1), F(2), \dots, F(n)).</math>
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 <math>\emptyset</math>.
 An <math>n</math>tuple, with <math>n > 0</math>, can be defined as an ordered pair of its first entry and an <math>(n1)</math>tuple (which contains the remaining entries when <math>n > 1</math>):
 <math>(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, a_3, \ldots, a_n))</math>
This definition can be applied recursively to the <math>(n1)</math>tuple:
 <math>(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, (a_3, (\ldots, (a_n, \emptyset)\ldots))))</math>
Thus, for example:
 <math>
\begin{align} (1, 2, 3) & = (1, (2, (3, \emptyset))) \\ (1, 2, 3, 4) & = (1, (2, (3, (4, \emptyset)))) \\ \end{align} </math>
A variant of this definition starts "peeling off" elements from the other end:
 The 0tuple is the empty set <math>\emptyset</math>.
 For <math>n > 0</math>:
 <math>(a_1, a_2, a_3, \ldots, a_n) = ((a_1, a_2, a_3, \ldots, a_{n1}), a_n)</math>
This definition can be applied recursively:
 <math>(a_1, a_2, a_3, \ldots, a_n) = ((\ldots(((\emptyset, a_1), a_2), a_3), \ldots), a_n)</math>
Thus, for example:
 <math>
\begin{align} (1, 2, 3) & = (((\emptyset, 1), 2), 3) \\ (1, 2, 3, 4) & = ((((\emptyset, 1), 2), 3), 4) \\ \end{align} </math>
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 <math>\emptyset</math>;
 Let <math>x</math> be an <math>n</math>tuple <math>(a_1, a_2, \ldots, a_n)</math>, and let <math>x \rightarrow b \equiv (a_1, a_2, \ldots, a_n, b)</math>. Then, <math>x \rightarrow b \equiv \{\{x\}, \{x, b\}\}</math>. (The right arrow, <math>\rightarrow</math>, could be read as "adjoined with".)
In this formulation:
 <math>
\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} </math>
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.^{[4]} 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.^{[5]} 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:
 <math>(x_1, x_2, \ldots, x_n) : \mathsf{T}_1 \times \mathsf{T}_2 \times \ldots \times \mathsf{T}_n</math>
and the projections are term constructors:
 <math>\pi_1(x) : \mathsf{T}_1,~\pi_2(x) : \mathsf{T}_2,~\ldots,~\pi_n(x) : \mathsf{T}_n</math>
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.^{[6]}
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 <math>S_1, S_2, \ldots, S_n</math> (note: the use of italics here that distinguishes sets from types) such that:
 <math>[\![\mathsf{T}_1]\!] = S_1,~[\![\mathsf{T}_2]\!] = S_2,~\ldots,~[\![\mathsf{T}_n]\!] = S_n</math>
and the interpretation of the basic terms is:
 <math>[\![x_1]\!] \in [\![\mathsf{T}_1]\!],~[\![x_2]\!] \in [\![\mathsf{T}_2]\!],~\ldots,~[\![x_n]\!] \in [\![\mathsf{T}_n]\!]</math>.
The <math>n</math>tuple of type theory has the natural interpretation as an <math>n</math>tuple of set theory:^{[7]}
 <math>[\![(x_1, x_2, \ldots, x_n)]\!] = (\,[\![x_1]\!], [\![x_2]\!], \ldots, [\![x_n]\!]\,)</math>
The unit type has as semantic interpretation the 0tuple.
See also
40x40px  Look up tuple in Wiktionary, the free dictionary. 
 Arity
 Exponential object
 Formal language
 OLAP: Multidimensional Expressions
 Prime ktuple
 Relation (mathematics)
 Tuplespace
Notes
 ^ "N‐tuple  Oxford Reference". oxfordreference.com. Retrieved 1 May 2015.
 ^ "Ordered ntuple  Oxford Reference". oxfordreference.com. Retrieved 1 May 2015.
 ^ 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
References
 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 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. 182–193
