Tuple

In mathematics, a tuple is a sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An $n$ -tuple is a tuple of $n$ elements, where $n$ is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called respectively a singleton and an ordered pair.

Tuple may be formally defined from ordered pairs by recurrence by starting from ordered pairs; indeed, a $n$ -tuple can be identified with the ordered pair of its $(n - 1)$ first elements and its $n$ th element.

Tuples are usually written by listing the elements within parentheses " $( )$ ", separated by a comma and a space; 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]

Etymology

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" (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".^[6]^[a]

Properties

The general rule for the identity of two $n$ -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:

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 $\{1,2,3\}=\{3,2,1\}$ .
If we have a tuple of $n$ elements $A$ then it is also a tuple of $i$ elements $A_{i}$ for each $i$ less than $n$ which contains first $i$ elements of original tuple $A$ .

The similarity of tuple to a set:

Tuple has intuitive count of elements, like a Set. However, only some definitions can have count defined from it's Set properties directly.
Tuple, like sets, can have infinite number of elements. Sequence is example of infinite number well ordered file having all properties of tuple. Such a general concept relies on Ordinal numbers, if to be defined exactly.

Definitions

There are several definitions of tuples that give them the properties described in the previous section. But only the first one doesn't suffer unwanted side-effects and has axiomatic relevance.

Tuples as functions

The $0$ -tuple is identified with the empty function.

For ordinal number $n\geq 1$ , the $n$ -tuple $\left(a_{i}:i<n\right)$ is identified with the (surjective) function

F~:~n=\left\{i:i<n\right\}~\to ~\left\{a_{i}:i<n\right\}

with domain

\operatorname {domain} F=n=\left\{i:i<n\right\}=\left\{i\in Ordinal:0\leq i<n\right\}

and with codomain

\operatorname {codomain} F=\left\{a_{i}:i<n\right\},

that is defined at $i\in \operatorname {domain} F$ by

F(i):=a_{i}.

That is, $F$ is the function defined by

{\begin{alignedat}{3}0\;&\mapsto &&\;a_{0}\\\;&\;\;\vdots &&\;\\i\;&\mapsto &&\;a_{i}\\\;&\;\;\vdots &&\;\\\end{alignedat}}

for any $i<n$ ,

in which case the equality

\left(a_{i}:i<n\right)=\left(F(i):i<n\right)

necessarily holds.

Count of elements in a tuple is defined intuitively as count (cardinality) of defining function $F$ as set of assigning pairs or equivalently as cardinality or ordinality of its domain. This property of definition is missing in following potentially alternative definitions.

This definition works not only for natural $n$ but for more general ordinal $n$ , making tuples potentially infinite. That allows to include sequences and furthers to concept of tuple without loosing the essential.

Note that indexing of elements of tuple is intrinsing and nonessential. Only position of element in the tuple defines it's order within the tuple. The domain of function defining the tuple is also intrinsing. Essential property of that domain is it's well ordedring. Therefore, in following text, all tuples usually index elements starting from 1, which is not an issue, explicit indexing is irrelevant, only well-ordering of intrinsing indexing matters. However, those tuples which express the last element behind the comma are valid only for nonlimit ordinal number element count. Correct expression for infinite sequence specifying also it's limit value would look like $\left(1,1/2,1/3,\dots 0\right)$ where $\dots 0$ expresses element at position of limit ordinal number $\omega$ . Such a position doestn't have defined its immediate predecessor. (Any position behind the coma has immediate predecessor.) Complementary problem doesn't exist as any ordinal number has defined immediate successor.

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 $F$ can be defined as:

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.

The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$ .
An $n$ -tuple, 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 0-tuple is the empty set $\emptyset$ .
For $n > 0$ :
$(a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n-1}),a_{n})$

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{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 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$ ;
Let $x$ be an $n$ -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}}

$n$ -tuples of $m$ -sets

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 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 $m n$ . 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 $n$ -fold Cartesian power $S \times S \times \dots \times S$ . 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 $n$ -tuple of type theory has the natural interpretation as an $n$ -tuple 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 0-tuple.

Notes

^ Compare the etymology of ploidy, from the Greek for -fold.

References

^ "Algebraic data type - HaskellWiki". wiki.haskell.org.
^ "Destructuring assignment". MDN Web Docs. 18 April 2023.
^ "Does JavaScript Guarantee Object Property Order?". Stack Overflow.
^ "N‐tuple". N‐tuple - Oxford Reference. Oxford University Press. January 2007. ISBN 9780199202720. Retrieved 1 May 2015. {{cite book}}: |work= ignored (help)
^ 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 & West 2000, p. 9
^ D'Angelo & 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, preprint

Sources

D'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 Lévy, 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. 182–193

External links

The dictionary definition of tuple at Wiktionary

[7] Compare the etymology of ploidy, from the Greek for -fold.

[1] "Algebraic data type - HaskellWiki". wiki.haskell.org.

[2] "Destructuring assignment". MDN Web Docs. 18 April 2023.

[3] "Does JavaScript Guarantee Object Property Order?". Stack Overflow.

[4] "N‐tuple". N‐tuple - Oxford Reference. Oxford University Press. January 2007. ISBN 9780199202720. Retrieved 1 May 2015. {{cite book}}: |work= ignored (help)

[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. Retrieved 2017-06-30. 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"

[8] D'Angelo & West 2000, p. 9

[9] D'Angelo & West 2000, p. 101

[pierce2002-10] Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. pp. 126–132. ISBN 0-262-16209-1.

[11] Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint

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v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo