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Wikipedia

This article is about the mathematical concept. For the musical term, see Tuplet.
"Octuple" redirects here. For the type of rowing boat, see Octuple scull.
"Duodecuple" redirects here. For the term in music, see Twelve-tone technique.

In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) ofn elements, wheren is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. Ann-tuple 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,(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, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. 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; and in philosophy.

Contents

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 numbern 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".

Names for tuples of specific lengths

Further information: Numeral prefix
Tuple length, n {\displaystyle n} Name Alternative names
0 empty tuple null tuple / empty sequence / unit / none left
1 monuple single / singleton / monad
2 couple double / ordered pair / two-ple / twin / dual / duad / dyad / twosome
3 triple treble / triplet / triad / ordered triple / threesome
4 quadruple quad / tetrad / quartet / quadruplet
5 quintuple pentuple / quint / pentad
6 sextuple hextuple / hexad
7 septuple heptuple / heptad
8 octuple octa / octet / octad / octuplet
9 nonuple nonad / ennead
10 decuple decad / decade (antiquated)
11 undecuple hendecuple / hendecad
12 duodecuple dozen / duodecad
13 tredecuple baker's dozen
14 quattuordecuple double septuple
15 quindecuple triple quintuple
16 sexdecuple quadruple quadruple
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
32 duotrigintuple
33 tritrigintuple
40 quadragintuple
41 unquadragintuple
50 quinquagintuple
60 sexagintuple
70 septuagintuple
80 octogintuple
90 nongentuple
100 centuple
1,000 milluple chiliad

Note that for n 3 {\displaystyle n\geq 3} , the tuple name in the table above can also function as a verb meaning "to multiply [the direct object] by n {\displaystyle n} "; for example, "to quintuple" means "to multiply by 5". If n = 2 {\displaystyle n=2} , then the associated verb is "to double". There is also a verb "sesquiple", meaning "to multiply by 3/2". Theoretically, "monuple" could be used in this way too.

The general rule for the identity of twon-tuples is

( a 1 , a 2 , , a n ) = ( b 1 , b 2 , , b n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})} if and only if a 1 = b 1 , a 2 = b 2 , , a n = b n {\displaystyle 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 ) ( 1 , 2 , 3 ) {\displaystyle (1,2,2,3)\neq (1,2,3)} ; but set { 1 , 2 , 2 , 3 } = { 1 , 2 , 3 } {\displaystyle \{1,2,2,3\}=\{1,2,3\}} .
  2. Tuple elements are ordered: tuple ( 1 , 2 , 3 ) ( 3 , 2 , 1 ) {\displaystyle (1,2,3)\neq (3,2,1)} , but set { 1 , 2 , 3 } = { 3 , 2 , 1 } {\displaystyle \{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.

There are several definitions of tuples that give them the properties described in the previous section.

Tuples as functions

The 0 {\displaystyle 0} -tuple may be identified as the empty function. For n 1 , {\displaystyle n\geq 1,} the n {\displaystyle n} -tuple ( a 1 , , a n ) {\displaystyle \left(a_{1},\ldots ,a_{n}\right)} may be identified with the (surjective) function

F : { 1 , , n } { a 1 , , a n } {\displaystyle F~:~\left\{1,\ldots ,n\right\}~\to ~\left\{a_{1},\ldots ,a_{n}\right\}}

with domain

domain F = { 1 , , n } = { i N : 1 i n } {\displaystyle \operatorname {domain} F=\left\{1,\ldots ,n\right\}=\left\{i\in \mathbb {N} :1\leq i\leq n\right\}}

and with codomain

codomain F = { a 1 , , a n } , {\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\},}

that is defined at i domain F = { 1 , , n } {\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}} by

F ( i ) := a i . {\displaystyle F(i):=a_{i}.}

That is, F {\displaystyle F} is the function defined by

1 a 1 n a n {\displaystyle {\begin{alignedat}{3}1\;&\mapsto &&\;a_{1}\\\;&\;\;\vdots &&\;\\n\;&\mapsto &&\;a_{n}\\\end{alignedat}}}

in which case the equality

( a 1 , a 2 , , a n ) = ( F ( 1 ) , F ( 2 ) , , F ( n ) ) {\displaystyle \left(a_{1},a_{2},\dots ,a_{n}\right)=\left(F(1),F(2),\dots ,F(n)\right)}

necessarily holds.

Tuples as sets of ordered pairs

Functions are commonly identified with their graphs, which is a certain set of ordered pairs. Indeed, many authors use graphs as the definition of a function. Using this definition of "function", the above function F {\displaystyle F} can be defined as:

F := { ( 1 , a 1 ) , , ( n , a n ) } . {\displaystyle F~:=~\left\{\left(1,a_{1}\right),\ldots ,\left(n,a_{n}\right)\right\}.}

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.

  1. The 0-tuple (i.e. the empty tuple) is represented by the empty set {\displaystyle \emptyset } .
  2. Ann-tuple, withn > 0, can be defined as an ordered pair of its first entry and an(n − 1)-tuple (which contains the remaining entries whenn > 1):
    ( a 1 , a 2 , a 3 , , a n ) = ( a 1 , ( a 2 , a 3 , , a n ) ) {\displaystyle (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 , , a n ) = ( a 1 , ( a 2 , ( a 3 , ( , ( a n , ) ) ) ) ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))}

Thus, for example:

( 1 , 2 , 3 ) = ( 1 , ( 2 , ( 3 , ) ) ) ( 1 , 2 , 3 , 4 ) = ( 1 , ( 2 , ( 3 , ( 4 , ) ) ) ) {\displaystyle {\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:

  1. The 0-tuple is the empty set {\displaystyle \emptyset } .
  2. Forn > 0:
    ( a 1 , a 2 , a 3 , , a n ) = ( ( a 1 , a 2 , a 3 , , a n 1 ) , a n ) {\displaystyle (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 , , a n ) = ( ( ( ( ( , a 1 ) , a 2 ) , a 3 ) , ) , a n ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})}

Thus, for example:

( 1 , 2 , 3 ) = ( ( ( , 1 ) , 2 ) , 3 ) ( 1 , 2 , 3 , 4 ) = ( ( ( ( , 1 ) , 2 ) , 3 ) , 4 ) {\displaystyle {\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:

  1. The 0-tuple (i.e. the empty tuple) is represented by the empty set {\displaystyle \emptyset } ;
  2. Let x {\displaystyle x} be ann-tuple ( a 1 , a 2 , , a n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})} , and let x b ( a 1 , a 2 , , a n , b ) {\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)} . Then, x b { { x } , { x , b } } {\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}} . (The right arrow, {\displaystyle \rightarrow } , could be read as "adjoined with".)

In this formulation:

( ) = ( 1 ) = ( ) 1 = { { ( ) } , { ( ) , 1 } } = { { } , { , 1 } } ( 1 , 2 ) = ( 1 ) 2 = { { ( 1 ) } , { ( 1 ) , 2 } } = { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } ( 1 , 2 , 3 ) = ( 1 , 2 ) 3 = { { ( 1 , 2 ) } , { ( 1 , 2 ) , 3 } } = { { { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } } , { { { { { } , { , 1 } } } , { { { } , { , 1 } } , 2 } } , 3 } } {\displaystyle {\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}}}

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 lengthn.n-tuples whose entries come from a set ofm elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number ofn-tuples of anm-set ismn. This follows from the combinatorial rule of product. IfS is a finite set of cardinalitym, this number is the cardinality of then-fold Cartesian powerS × S × ⋯ × S. Tuples are elements of this product set.

Main article: Product type

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 , , x n ) : T 1 × T 2 × × T n {\displaystyle (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:

π 1 ( x ) : T 1 , π 2 ( x ) : T 2 , , π n ( x ) : T n {\displaystyle \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.

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 , , S n {\displaystyle S_{1},S_{2},\ldots ,S_{n}} (note: the use of italics here that distinguishes sets from types) such that:

[ [ T 1 ] ] = S 1 , [ [ T 2 ] ] = S 2 , , [ [ T n ] ] = S n {\displaystyle [\![{\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 ] ] [ [ T 1 ] ] , [ [ x 2 ] ] [ [ T 2 ] ] , , [ [ x n ] ] [ [ T n ] ] {\displaystyle [\![x_{1}]\!]\in [\![{\mathsf {T}}_{1}]\!],~[\![x_{2}]\!]\in [\![{\mathsf {T}}_{2}]\!],~\ldots ,~[\![x_{n}]\!]\in [\![{\mathsf {T}}_{n}]\!]} .

Then-tuple of type theory has the natural interpretation as ann-tuple of set theory:

[ [ ( x 1 , x 2 , , x n ) ] ] = ( [ [ x 1 ] ] , [ [ x 2 ] ] , , [ [ x n ] ] ) {\displaystyle [\![(x_{1},x_{2},\ldots ,x_{n})]\!]=(\,[\![x_{1}]\!],[\![x_{2}]\!],\ldots ,[\![x_{n}]\!]\,)}

The unit type has as semantic interpretation the 0-tuple.

  1. Compare the etymology of ploidy, from the Greek for -fold.
  1. "Algebraic data type - HaskellWiki". wiki.haskell.org.
  2. "Destructuring assignment". MDN Web Docs.
  3. "Does JavaScript Guarantee Object Property Order?". Stack Overflow.
  4. "N‐tuple". N‐tuple - Oxford Reference. oxfordreference.com. Oxford University Press. January 2007. ISBN 9780199202720. Retrieved1 May 2015.
  5. Blackburn, Simon (1994). "ordered n-tuple". The Oxford Dictionary of Philosophy. Oxford guidelines quick reference (3 ed.). Oxford: Oxford University Press (published 2016). p. 342. ISBN 9780198735304. Retrieved2017-06-30. ordered n-tuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.
  6. OED, s.v. "triple", "quadruple", "quintuple", "decuple"
  7. D'Angelo & West 2000, p. 9
  8. D'Angelo & West 2000, p. 101
  9. Pierce, Benjamin (2002).Types and Programming Languages. MIT Press. pp. 126–132. ISBN 0-262-16209-1.
  10. Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint
  • The dictionary definition of tuple at Wiktionary

Tuple Article Talk Language Watch Edit This article is about the mathematical concept For the musical term see Tuplet Octuple redirects here For the type of rowing boat see Octuple scull Duodecuple redirects here For the term in music see Twelve tone technique In mathematics a tuple is a finite ordered list sequence of elements An n tuple is a sequence or ordered list of n elements where n is a non negative integer There is only one 0 tuple referred to as the empty tuple An n tuple 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 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 Contents 1 Etymology 1 1 Names for tuples of specific lengths 2 Properties 3 Definitions 3 1 Tuples as functions 3 2 Tuples as nested ordered pairs 3 3 Tuples as nested sets 4 n tuples of m sets 5 Type theory 6 See also 7 Notes 8 References 9 Sources 10 External linksEtymology EditThe 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 ploῦs which replaced the classical and late antique plex meaning folded as in duplex 6 a Names for tuples of specific lengths Edit Further information Numeral prefix Tuple length n displaystyle n Name Alternative names0 empty tuple null tuple empty sequence unit none left1 monuple single singleton monad2 couple double ordered pair two ple twin dual duad dyad twosome3 triple treble triplet triad ordered triple threesome4 quadruple quad tetrad quartet quadruplet5 quintuple pentuple quint pentad6 sextuple hextuple hexad7 septuple heptuple heptad8 octuple octa octet octad octuplet9 nonuple nonad ennead10 decuple decad decade antiquated 11 undecuple hendecuple hendecad12 duodecuple dozen duodecad13 tredecuple baker s dozen14 quattuordecuple double septuple15 quindecuple triple quintuple16 sexdecuple quadruple quadruple17 septendecuple18 octodecuple19 novemdecuple20 vigintuple21 unvigintuple22 duovigintuple23 trevigintuple24 quattuorvigintuple25 quinvigintuple26 sexvigintuple27 septenvigintuple28 octovigintuple29 novemvigintuple30 trigintuple31 untrigintuple32 duotrigintuple33 tritrigintuple40 quadragintuple41 unquadragintuple50 quinquagintuple60 sexagintuple70 septuagintuple80 octogintuple90 nongentuple100 centuple1 000 milluple chiliad Note that for n 3 displaystyle n geq 3 the tuple name in the table above can also function as a verb meaning to multiply the direct object by n displaystyle n for example to quintuple means to multiply by 5 If n 2 displaystyle n 2 then the associated verb is to double There is also a verb sesquiple meaning to multiply by 3 2 Theoretically monuple could be used in this way too Properties EditThe general rule for the identity of two n tuples is a 1 a 2 a n b 1 b 2 b n displaystyle a 1 a 2 ldots a n b 1 b 2 ldots b n if and only if a 1 b 1 a 2 b 2 a n b n displaystyle 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 1 2 3 displaystyle 1 2 2 3 neq 1 2 3 but set 1 2 2 3 1 2 3 displaystyle 1 2 2 3 1 2 3 Tuple elements are ordered tuple 1 2 3 3 2 1 displaystyle 1 2 3 neq 3 2 1 but set 1 2 3 3 2 1 displaystyle 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 EditThere are several definitions of tuples that give them the properties described in the previous section Tuples as functions Edit The 0 displaystyle 0 tuple may be identified as the empty function For n 1 displaystyle n geq 1 the n displaystyle n tuple a 1 a n displaystyle left a 1 ldots a n right may be identified with the surjective function F 1 n a 1 a n displaystyle F left 1 ldots n right to left a 1 ldots a n right with domain domain F 1 n i N 1 i n displaystyle operatorname domain F left 1 ldots n right left i in mathbb N 1 leq i leq n right and with codomain codomain F a 1 a n displaystyle operatorname codomain F left a 1 ldots a n right that is defined at i domain F 1 n displaystyle i in operatorname domain F left 1 ldots n right by F i a i displaystyle F i a i That is F displaystyle F is the function defined by 1 a 1 n a n displaystyle begin alignedat 3 1 amp mapsto amp amp a 1 amp vdots amp amp n amp mapsto amp amp a n end alignedat in which case the equality a 1 a 2 a n F 1 F 2 F n displaystyle left a 1 a 2 dots a n right left F 1 F 2 dots F n right necessarily holds Tuples as sets of ordered pairs Functions are commonly identified with their graphs which is a certain set of ordered pairs Indeed many authors use graphs as the definition of a function Using this definition of function the above function F displaystyle F can be defined as F 1 a 1 n a n displaystyle F left left 1 a 1 right ldots left n a n right right Tuples as nested ordered pairs Edit 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 The 0 tuple i e the empty tuple is represented by the empty set displaystyle emptyset An n tuple with n gt 0 can be defined as an ordered pair of its first entry and an n 1 tuple which contains the remaining entries when n gt 1 a 1 a 2 a 3 a n a 1 a 2 a 3 a n displaystyle 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 a n a 1 a 2 a 3 a n displaystyle a 1 a 2 a 3 ldots a n a 1 a 2 a 3 ldots a n emptyset ldots Thus for example 1 2 3 1 2 3 1 2 3 4 1 2 3 4 displaystyle begin aligned 1 2 3 amp 1 2 3 emptyset 1 2 3 4 amp 1 2 3 4 emptyset end aligned A variant of this definition starts peeling off elements from the other end The 0 tuple is the empty set displaystyle emptyset For n gt 0 a 1 a 2 a 3 a n a 1 a 2 a 3 a n 1 a n displaystyle 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 a n a 1 a 2 a 3 a n displaystyle a 1 a 2 a 3 ldots a n ldots emptyset a 1 a 2 a 3 ldots a n Thus for example 1 2 3 1 2 3 1 2 3 4 1 2 3 4 displaystyle begin aligned 1 2 3 amp emptyset 1 2 3 1 2 3 4 amp emptyset 1 2 3 4 end aligned Tuples as nested sets Edit Using Kuratowski s representation for an ordered pair the second definition above can be reformulated in terms of pure set theory The 0 tuple i e the empty tuple is represented by the empty set displaystyle emptyset Let x displaystyle x be an n tuple a 1 a 2 a n displaystyle a 1 a 2 ldots a n and let x b a 1 a 2 a n b displaystyle x rightarrow b equiv a 1 a 2 ldots a n b Then x b x x b displaystyle x rightarrow b equiv x x b The right arrow displaystyle rightarrow could be read as adjoined with In this formulation 1 1 1 1 1 2 1 2 1 1 2 1 1 2 1 2 3 1 2 3 1 2 1 2 3 1 1 2 1 1 2 3 displaystyle begin array lclcl amp amp amp amp emptyset amp amp amp amp 1 amp amp rightarrow 1 amp amp 1 amp amp amp amp emptyset emptyset 1 amp amp amp amp 1 2 amp amp 1 rightarrow 2 amp amp 1 1 2 amp amp amp amp emptyset emptyset 1 amp amp amp amp emptyset emptyset 1 2 amp amp amp amp 1 2 3 amp amp 1 2 rightarrow 3 amp amp 1 2 1 2 3 amp amp amp amp emptyset emptyset 1 amp amp amp amp emptyset emptyset 1 2 amp amp amp amp emptyset emptyset 1 amp amp amp amp emptyset emptyset 1 2 3 end array n tuples of m sets EditIn 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 Type theory EditMain article Product type 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 x n T 1 T 2 T n displaystyle 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 p 1 x T 1 p 2 x T 2 p n x T n displaystyle 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 S n displaystyle S 1 S 2 ldots S n note the use of italics here that distinguishes sets from types such that T 1 S 1 T 2 S 2 T n S n displaystyle 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 T 1 x 2 T 2 x n T n displaystyle x 1 in mathsf T 1 x 2 in mathsf T 2 ldots x n in mathsf T n The n tuple of type theory has the natural interpretation as an n tuple of set theory 10 x 1 x 2 x n x 1 x 2 x n displaystyle x 1 x 2 ldots x n x 1 x 2 ldots x n The unit type has as semantic interpretation the 0 tuple See also EditArity Coordinate vector Exponential object Formal language OLAP Multidimensional Expressions Prime k tuple Relation mathematics Sequence TuplespaceNotes Edit Compare the etymology of ploidy from the Greek for fold References Edit Algebraic data type HaskellWiki wiki haskell org Destructuring assignment MDN Web Docs Does JavaScript Guarantee Object Property Order Stack Overflow N tuple N tuple Oxford Reference oxfordreference com Oxford University Press January 2007 ISBN 9780199202720 Retrieved 1 May 2015 Blackburn Simon 1994 ordered n tuple The Oxford Dictionary of Philosophy Oxford guidelines quick reference 3 ed Oxford Oxford University Press published 2016 p 342 ISBN 9780198735304 Retrieved 2017 06 30 ordered n tuple A generalization of the notion of an ordered pair to sequences of n objects OED s v triple quadruple quintuple decuple D Angelo amp West 2000 p 9 D Angelo amp West 2000 p 101 Pierce Benjamin 2002 Types and Programming Languages MIT Press pp 126 132 ISBN 0 262 16209 1 Steve Awodey From sets to types to categories to sets 2009 preprintSources EditD Angelo John P West Douglas B 2000 Mathematical Thinking Problem Solving and Proofs 2nd ed Prentice Hall ISBN 978 0 13 014412 6 Keith Devlin The Joy of Sets Springer Verlag 2nd ed 1993 ISBN 0 387 94094 4 pp 7 8 Abraham Adolf Fraenkel Yehoshua Bar Hillel Azriel Levy Foundations of school Set Theory Elsevier Studies in Logic Vol 67 2nd Edition revised 1973 ISBN 0 7204 2270 1 p 33 Gaisi Takeuti W M Zaring Introduction to Axiomatic Set Theory Springer GTM 1 1971 ISBN 978 0 387 90024 7 p 14 George J Tourlakis Lecture Notes in Logic and Set Theory Volume 2 Set Theory Cambridge University Press 2003 ISBN 978 0 521 75374 6 pp 182 193External links Edit The dictionary definition of tuple at Wiktionary Retrieved from https en wikipedia org w index php title Tuple amp oldid 1090976042, wikipedia, wiki, book,

books

, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.