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{{About|the mathematical concept|the musical term|Tuplet}}{{Redir|Octuple|the type of rowing boat|Octuple scull}}{{Redir|Duodecuple|the term in music|Twelve-tone technique}}In mathematics, a tuple is a finite ordered list (sequence) of elements. An {{math|n}}-tuple is a sequence (or ordered list) of {{math|n}} elements, where {{math|n}} is a non-negative integer. There is only one 0-tuple, an empty sequence, or empty tuple, as it is referred to. An {{math|n}}-tuple 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 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "âŸ¨ âŸ©". 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|date =May 2017}} Most typed functional programming languages implement tuples directly as product types,WEB,weblink Algebraic data type - HaskellWiki, wiki.haskell.org, tightly associated with algebraic data types, pattern matching, and destructuring assignment.WEB,weblink Destructuring assignment, MDN Web Docs, Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label.WEB,weblink Does JavaScript Guarantee Object Property Order?, Stack Overflow, 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;WEB,weblink Nâ€tuple - Oxford Reference, oxfordreference.com, 1 May 2015, and in philosophy.BOOK, Blackburn, Simon, Simon Blackburn, 1994, ordered n-tuple, The Oxford Dictionary of Philosophy,weblink 3, Oxford, Oxford University Press, 2016, 342, Oxford quick reference, 9780198735304, 2017-06-30, ordered n-tuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.,

## Etymology

The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., {{math|n}}â€‘tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0â€‘tuple is called the null tuple. A 1â€‘tuple is called a singleton, a 2â€‘tuple is called an ordered pair or couple, and a 3â€‘tuple is called a triple or triplet. The number {{math|n}} can be any nonnegative integer. For example, a complex number can be represented as a 2â€‘tuple of reals, a quaternion can be represented as a 4â€‘tuple, an octonion can be represented as an 8â€‘tuple, and a sedenion can be represented as a 16â€‘tuple.Although these uses treat â€‘tuple as the suffix, the original suffix was â€‘ple as in "triple" (three-fold) or "decuple" (tenâ€‘fold). This originates from medieval Latin plus (meaning "more") related to Greek â€‘Ï€Î»Î¿á¿¦Ï‚, which replaced the classical and late antique â€‘plex (meaning "folded"), as in "duplex".OED, s.v. "triple", "quadruple", "quintuple", "decuple"{{efn|Compare the etymology of ploidy, from the Greek for -fold.}}

### Names for tuples of specific lengths

{{cot|Table of names and variants for specific lengths}}{| class="wikitable"! Tuple length, n !! Name !! Alternative names
0 empty tuple unit / empty sequence / null tuple
1 monuple singleton / single / monad
2 couple dual / double / (ordered) pair / twin / duad / two-ple
3 triple treble / triplet / triad
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
50 quinquagintuple
60 sexagintuple
70 septuagintuple
80 octogintuple
90 nongentuple
100 centuple
1,000 milluple
{{cob}}

## Properties

The general rule for the identity of two {{math|n}}-tuples 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.
1. 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}.
2. Tuple elements are ordered: tuple (1,2,3) neq (3,2,1), but set {1,2,3} = {3,2,1}.
3. 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}}-tuple can be regarded as a function, {{math|F}}, whose domain is the tuple's implicit set of element indices, {{math|X}}, and whose codomain, {{math|Y}}, is the tuple's set of elements. Formally:
(a_1, a_2, dots, a_n) equiv (X,Y,F)
where:
begin{align}
X & = {1, 2, dots, n} = {iinmathbb Nmid 1leq ileq n}
Y & = {a_1, a_2, ldots, a_n}
F & = {(1, a_1), (2, a_2), ldots, (n, a_n)}.
end{align}
In slightly less formal notation this says:
(a_1, a_2, dots, a_n) := (F(1), F(2), dots, F(n)).
Using this definition of n-tuples, it follows that there is only one 0-tuple, the empty function.

### 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 2-tuple
1. The 0-tuple (i.e. the empty tuple) is represented by the empty set emptyset.
2. An {{math|n}}-tuple, with {{math|n > 0}}, can be defined as an ordered pair of its first entry and an {{math|(n âˆ’ 1)}}-tuple (which contains the remaining entries when {{math|n > 1)}}:
3. : (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 {{math|(n âˆ’ 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{align}
(1, 2, 3) & = (1, (2, (3, emptyset)))
(1, 2, 3, 4) & = (1, (2, (3, (4, emptyset))))
end{align}
A variant of this definition starts "peeling off" elements from the other end:
1. The 0-tuple is the empty set emptyset.
2. For {{math|n > 0}}:
3. : (a_1, a_2, a_3, ldots, a_n) = ((a_1, a_2, a_3, ldots, a_{n-1}), 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{align}
(1, 2, 3) & = (((emptyset, 1), 2), 3)
(1, 2, 3, 4) & = ((((emptyset, 1), 2), 3), 4)
end{align}

### 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:
1. The 0-tuple (i.e. the empty tuple) is represented by the empty set emptyset;
2. Let x be an {{math|n}}-tuple (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}

## {{anchor|n-tuple}}{{math|n}}-tuples of {{math|m}}-sets

In discrete mathematics, especially combinatorics and finite probability theory, {{math|n}}-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length {{math|n}}.{{harvnb|D'Angelo|West|2000|p=9}} {{math|n}}-tuples whose entries come from a set of {{math|m}} elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of {{math|n}}-tuples of an {{math|m}}-set is {{math|mn}}. This follows from the combinatorial rule of product.{{harvnb|D'Angelo|West|2000|p=101}} If {{math|S}} is a finite set of cardinality {{math|m}}, this number is the cardinality of the {{math|n}}-fold Cartesian power {{math|S Ã— S Ã— ... 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.BOOK, Pierce, Benjamin, Types and Programming Languages, MIT Press, 2002, 0-262-16209-1, 126â€“132, 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 {{math|n}}-tuple of type theory has the natural interpretation as an {{math|n}}-tuple of set theory:Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint
[![(x_1, x_2, ldots, x_n)]!] = (,[![x_1]!], [![x_2]!], ldots, [![x_n]!],)
The unit type has as semantic interpretation the 0-tuple.

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