# Tuple

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Tuple
> Markdown URL: https://mediated.wiki/source/Tuple.md
> Source: https://en.wikipedia.org/wiki/Tuple
> Source revision: 1345618352
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Finite ordered list of elements

For the musical term, see [Tuplet](/source/Tuplet).

"Octuple" redirects here. For the boat, see [Octuple scull](/source/Octuple_scull).

"Sextuple" redirects here. For the sports achievement in association football, see [Sextuple (association football)](/source/Sextuple_(association_football)).

In [mathematics](/source/Mathematics), a **tuple** is a finite [sequence](/source/Sequence) (or ordered list) of [numbers](/source/Number). More generally, it is a sequence of [mathematical objects](/source/Mathematical_object), called the *elements* of the tuple. An **n-tuple** is a tuple of n elements, where n is a non-negative [integer](/source/Integer). There is only one 0-tuple, called the *empty tuple*. A 1-tuple and a 2-tuple are commonly called a [*singleton*](/source/Singleton_(mathematics)) and an *[ordered pair](/source/Ordered_pair)*, respectively. The term *"infinite tuple"* is occasionally used for *"infinite sequences"*.

Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning.[a]

An n-tuple can be formally defined as the [image](/source/Image_(mathematics)) of a [function](/source/Function_(mathematics)) that has the set of the first n [natural numbers](/source/Natural_number) as its [domain](/source/Domain_of_a_function) (1, 2, ..., *n*). Tuples may be also defined from ordered pairs by a [recurrence](/source/Recurrence_relation) starting from an ordered pair; indeed, an n-tuple can be identified with the ordered pair of its (*n* − 1) first elements and its nth element, for example, ( ( ( 1 , 2 ) , 3 ) , 4 ) = ( 1 , 2 , 3 , 4 ) {\displaystyle \left(\left(\left(1,2\right),3\right),4\right)=\left(1,2,3,4\right)} .

In [computer science](/source/Computer_science), tuples come in many forms. Most typed [functional programming](/source/Functional_programming) languages implement tuples directly as [product types](/source/Product_type),[1] tightly associated with [algebraic data types](/source/Algebraic_data_type), [pattern matching](/source/Pattern_matching), and [destructuring assignment](/source/Assignment_(computer_science)#Parallel_assignment).[2] Many programming languages offer an alternative to tuples, known as [record types](/source/Record_(computer_science)), 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](/source/Struct_(C_programming_language)) and Haskell records. [Relational databases](/source/Relational_database) may formally identify their [rows](/source/Row_(database)) (records) as *tuples*.

Tuples also occur in [relational algebra](/source/Relational_algebra); when programming the [semantic web](/source/Semantic_web) with the [Resource Description Framework](/source/Resource_Description_Framework) (RDF); in [linguistics](/source/Linguistics);[4] and in [philosophy](/source/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](/source/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](/source/Integer). For example, a [complex number](/source/Complex_number) can be represented as a 2‑tuple of reals, a [quaternion](/source/Quaternion) can be represented as a 4‑tuple, an [octonion](/source/Octonion) can be represented as an 8‑tuple, and a [sedenion](/source/Sedenion) can be represented as a 16‑tuple.

Although these uses treat *‑tuple* as the suffix, the original suffix was *‑ple* as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from [medieval Latin](/source/Medieval_Latin) *plus* (meaning "more") related to [Greek](/source/Greek_language) ‑πλοῦς, which replaced the classical and late antique *‑plex* (meaning "folded"), as in "duplex".[6][b]

## Properties

The 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](/source/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](/source/Set_(mathematics)):

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\}} .

1. 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\}} .

1. A tuple has a finite number of elements, while a set or a [multiset](/source/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

The 0 {\displaystyle 0} -tuple may be identified as the [empty function](/source/Function_(mathematics)#General_properties). 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](/source/Surjective_function) [function](/source/Function_(mathematics)#Definition)

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

with [domain](/source/Domain_of_a_function)

- 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](/source/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](/source/Graph_of_a_function), 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](/source/Ordered_pair). 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 } .

1. 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): 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 } .

1. For *n* > 0: 1. ( 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](/source/Ordered_pair#Kuratowski's_definition), the second definition above can be reformulated in terms of pure [set theory](/source/Set_theory):

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ∅ {\displaystyle \emptyset } ;

1. 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}()&&&=&\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](/source/Discrete_mathematics), especially [combinatorics](/source/Combinatorics) and finite [probability theory](/source/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](/source/Permutation#Permutations_of_multisets)* 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](/source/Rule_of_product).[8] If *S* is a finite set of [cardinality](/source/Cardinality) *m*, this number is the cardinality of the *n*-fold [Cartesian power](/source/Cartesian_product#n-ary_Cartesian_power) *S* × *S* × ⋯ × *S*. Tuples are elements of this product set.

## Type theory

Main article: [Product type](/source/Product_type)

In [type theory](/source/Type_theory), commonly used in [programming languages](/source/Programming_language), a tuple has a [product type](/source/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](/source/Projection_(mathematics)) 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](/source/Relational_model) has a [record type](/source/Record_(computer_science)). Both of these types can be defined as simple extensions of the [simply typed lambda calculus](/source/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](/source/Model_theory) 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](/source/Unit_type) has as semantic interpretation the 0-tuple.

For a list of tuple types in programming languages, see [Product type#Product types in programming languages](/source/Product_type#Product_types_in_programming_languages).

## See also

- [Arity](/source/Arity)

- [Coordinate vector](/source/Coordinate_vector)

- [Exponential object](/source/Exponential_object)

- [Formal language](/source/Formal_language)

- [Multidimensional Expressions](/source/Multidimensional_Expressions#MDX_data_types) (OLAP)

- [Prime *k*-tuple](/source/Prime_k-tuple)

- [Relation (mathematics)](/source/Relation_(mathematics))

- [Sequence](/source/Sequence)

- [Tuplespace](/source/Tuplespace)

- [Tuple Names](https://simple.wikipedia.org/wiki/Tuple_names)

## Notes

1. **[^](#cite_ref-1)** [Square brackets](/source/Square_bracket) are used for [matrices](/source/Matrix_(mathematics)), including [row vectors](/source/Row_vector). [Braces](/source/Braces_(punctuation)) are used for [sets](/source/Set_(mathematics)). Each [programming language](/source/Programming_language) has its own convention for the different brackets.

1. **[^](#cite_ref-8)** Compare the etymology of [ploidy](/source/Ploidy), from the Greek for -fold.

## References

1. **[^](#cite_ref-2)** ["Algebraic data type - HaskellWiki"](https://wiki.haskell.org/Algebraic_data_type). *wiki.haskell.org*.

1. **[^](#cite_ref-3)** ["Destructuring assignment"](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/Destructuring_assignment). *MDN Web Docs*. 18 April 2023.

1. **[^](#cite_ref-4)** ["Does JavaScript Guarantee Object Property Order?"](https://stackoverflow.com/q/5525795). *Stack Overflow*.

1. **[^](#cite_ref-5)** Matthews, P. H., ed. (January 2007). ["N-tuple"](http://www.oxfordreference.com/view/10.1093/acref/9780199202720.001.0001/acref-9780199202720-e-2276). *The Concise Oxford Dictionary of Linguistics*. Oxford University Press. [ISBN](/source/ISBN_(identifier)) [9780199202720](https://en.wikipedia.org/wiki/Special:BookSources/9780199202720). Retrieved 1 May 2015.

1. **[^](#cite_ref-6)** [Blackburn, Simon](/source/Simon_Blackburn) (1994). "ordered n-tuple". [*The Oxford Dictionary of Philosophy*](https://books.google.com/books?id=Mno8CwAAQBAJ). Oxford guidelines quick reference (3 ed.). Oxford: Oxford University Press (published 2016). p. 342. [ISBN](/source/ISBN_(identifier)) [9780198735304](https://en.wikipedia.org/wiki/Special:BookSources/9780198735304). Retrieved 2017-06-30. ordered n-tuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.

1. **[^](#cite_ref-7)** *OED*, *s.v.* "triple", "quadruple", "quintuple", "decuple"

1. **[^](#cite_ref-9)** [D'Angelo & West 2000](#CITEREFD'AngeloWest2000), p. 9

1. **[^](#cite_ref-10)** [D'Angelo & West 2000](#CITEREFD'AngeloWest2000), p. 101

1. **[^](#cite_ref-pierce2002_11-0)** Pierce, Benjamin (2002). [*Types and Programming Languages*](https://archive.org/details/typesprogramming00pier_207). MIT Press. pp. [126](https://archive.org/details/typesprogramming00pier_207/page/n149)–132. [ISBN](/source/ISBN_(identifier)) [0-262-16209-1](https://en.wikipedia.org/wiki/Special:BookSources/0-262-16209-1).

1. **[^](#cite_ref-12)** Steve Awodey, [*From sets, to types, to categories, to sets*](http://www.andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf), 2009, [preprint](/source/Preprint)

## Sources

- [D'Angelo, John P.](/source/John_D'Angelo); West, Douglas B. (2000), *Mathematical Thinking/Problem-Solving and Proofs* (2nd ed.), Prentice-Hall, [ISBN](/source/ISBN_(identifier)) [978-0-13-014412-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-014412-6)

- [Keith Devlin](/source/Keith_Devlin), *The Joy of Sets*. Springer Verlag, 2nd ed., 1993, [ISBN](/source/ISBN_(identifier)) [0-387-94094-4](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94094-4), pp. 7–8

- [Abraham Adolf Fraenkel](/source/Abraham_Adolf_Fraenkel), [Yehoshua Bar-Hillel](/source/Yehoshua_Bar-Hillel), [Azriel Lévy](/source/Azriel_L%C3%A9vy), *[Foundations of school Set Theory](https://books.google.com/books?id=ah2bwOwc06MC)*, Elsevier Studies in Logic Vol. 67, 2nd Edition, revised, 1973, [ISBN](/source/ISBN_(identifier)) [0-7204-2270-1](https://en.wikipedia.org/wiki/Special:BookSources/0-7204-2270-1), p. 33

- [Gaisi Takeuti](/source/Gaisi_Takeuti), W. M. Zaring, *Introduction to Axiomatic Set Theory*, Springer [GTM](/source/Graduate_Texts_in_Mathematics) 1, 1971, [ISBN](/source/ISBN_(identifier)) [978-0-387-90024-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90024-7), p. 14

- George J. Tourlakis, *[Lecture Notes in Logic and Set Theory. Volume 2: Set Theory](https://books.google.com/books?as_isbn=9780521753746)*, Cambridge University Press, 2003, [ISBN](/source/ISBN_(identifier)) [978-0-521-75374-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-75374-6), pp. 182–193

## External links

- The dictionary definition of [*tuple*](https://en.wiktionary.org/wiki/tuple) at Wiktionary

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 Constructive 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

Authority control databases GND

---
Adapted from the Wikipedia article [Tuple](https://en.wikipedia.org/wiki/Tuple) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Tuple?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.