tuple

The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., {{math|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 {{math|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 ‑tuple 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".OED, s.v. "triple", "quadruple", "quintuple", "decuple"{{efn|Compare the etymology of ploidy, from the Greek for -fold.}}

The general rule for the identity of two {{math|n}}-tuples is

: $(a\_1,\; a\_2,\; \backslash ldots,\; a\_n)\; =\; (b\_1,\; b\_2,\; \backslash ldots,\; b\_n)$ if and only if $a\_1=b\_1,\backslash text\{\; \}a\_2=b\_2,\backslash text\{\; \}\backslash ldots,\backslash text\{\; \}a\_n=b\_n$.

Thus a tuple has properties that distinguish it from a set:

1. A tuple may contain multiple instances of the same element, so
   tuple $(1,2,2,3)\; \backslash neq\; (1,2,3)$; but set $\backslash \{1,2,2,3\backslash \}\; =\; \backslash \{1,2,3\backslash \}$.
2. Tuple elements are ordered: tuple $(1,2,3)\; \backslash neq\; (3,2,1)$, but set $\backslash \{1,2,3\backslash \}\; =\; \backslash \{3,2,1\backslash \}$.
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.

The $0$-tuple may be identified as the empty function. For $n\; \backslash geq\; 1,$ the $n$-tuple $\backslash left(a\_1,\; \backslash ldots,\; a\_n\backslash right)$ may be identified with the (surjective) function

:$F\; ~:~\; \backslash left\backslash \{\; 1,\; \backslash ldots,\; n\; \backslash right\backslash \}\; ~\backslash to~\; \backslash left\backslash \{\; a\_1,\; \backslash ldots,\; a\_n\; \backslash right\backslash \}$

with domain

:$\backslash operatorname\{domain\}\; F\; =\; \backslash left\backslash \{\; 1,\; \backslash ldots,\; n\; \backslash right\backslash \}\; =\; \backslash left\backslash \{\; i\; \backslash in\; \backslash N\; :\; 1\; \backslash leq\; i\; \backslash leq\; n\backslash right\backslash \}$

and with codomain

:$\backslash operatorname\{codomain\}\; F\; =\; \backslash left\backslash \{\; a\_1,\; \backslash ldots,\; a\_n\; \backslash right\backslash \},$

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

:$F(i)\; :=\; a\_i.$

That is, $F$ is the function defined by

:$\backslash begin\{alignat\}\{3\}$

1 \;&\mapsto&&\; a_1 \\

\;&\;\;\vdots&&\; \\

n \;&\mapsto&&\; a_n \\

\end{alignat}

in which case the equality

:$\backslash left(a\_1,\; a\_2,\; \backslash dots,\; a\_n\backslash right)\; =\; \backslash left(F(1),\; F(2),\; \backslash dots,\; F(n)\backslash right)$

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 $F$ can be defined as:

:$F\; ~:=~\; \backslash left\backslash \{\; \backslash left(1,\; a\_1\backslash right),\; \backslash ldots,\; \backslash left(n,\; a\_n\backslash right)\; \backslash right\backslash \}.$

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 $\backslash emptyset$.
2. An {{math|n}}-tuple, with {{math|n > 0}}, can be defined as an ordered pair of its first entry and an {{math|(n − 1)}}-tuple (which contains the remaining entries when {{math|n > 1)}}:
3. : $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; (a\_1,\; (a\_2,\; a\_3,\; \backslash ldots,\; a\_n))$

This definition can be applied recursively to the {{math|(n − 1)}}-tuple:

: $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; (a\_1,\; (a\_2,\; (a\_3,\; (\backslash ldots,\; (a\_n,\; \backslash emptyset)\backslash ldots))))$

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 $\backslash emptyset$.
2. For {{math|n > 0}}:
3. : $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; ((a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_\{n-1\}),\; a\_n)$

This definition can be applied recursively:

: $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; ((\backslash ldots(((\backslash emptyset,\; a\_1),\; a\_2),\; a\_3),\; \backslash ldots),\; a\_n)$

Thus, for example:

: $$

\begin{align}

(1, 2, 3) & = (((\emptyset, 1), 2), 3) \\

(1, 2, 3, 4) & = ((((\emptyset, 1), 2), 3), 4) \\

\end{align}

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 $\backslash emptyset$;
2. Let $x$ be an {{math|n}}-tuple $(a\_1,\; a\_2,\; \backslash ldots,\; a\_n)$, and let $x\; \backslash rightarrow\; b\; \backslash equiv\; (a\_1,\; a\_2,\; \backslash ldots,\; a\_n,\; b)$. Then, $x\; \backslash rightarrow\; b\; \backslash equiv\; \backslash \{\backslash \{x\backslash \},\; \backslash \{x,\; b\backslash \}\backslash \}$. (The right arrow, $\backslash rightarrow$, 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}

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 × S × ⋯ × S}}. Tuples are elements of this product set.

{{main|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:

: $(x\_1,\; x\_2,\; \backslash ldots,\; x\_n)\; :\; \backslash mathsf\{T\}\_1\; \backslash times\; \backslash mathsf\{T\}\_2\; \backslash times\; \backslash ldots\; \backslash times\; \backslash mathsf\{T\}\_n$

and the projections are term constructors:

: $\backslash pi\_1(x)\; :\; \backslash mathsf\{T\}\_1,~\backslash pi\_2(x)\; :\; \backslash mathsf\{T\}\_2,~\backslash ldots,~\backslash pi\_n(x)\; :\; \backslash mathsf\{T\}\_n$

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.{{cite book|last=Pierce|first=Benjamin|title=Types and Programming Languages|url=https://archive.org/details/typesprogramming00pier_207|url-access=limited|publisher=MIT Press|year=2002|isbn=0-262-16209-1|pages=[https://archive.org/details/typesprogramming00pier_207/page/n149 126]–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 $S\_1,\; S\_2,\; \backslash ldots,\; S\_n$ (note: the use of italics here that distinguishes sets from types) such that:

: $[\backslash ![\backslash mathsf\{T\}\_1]\backslash !]\; =\; S\_1,~[\backslash ![\backslash mathsf\{T\}\_2]\backslash !]\; =\; S\_2,~\backslash ldots,~[\backslash ![\backslash mathsf\{T\}\_n]\backslash !]\; =\; S\_n$

and the interpretation of the basic terms is:

: $[\backslash ![x\_1]\backslash !]\; \backslash in\; [\backslash ![\backslash mathsf\{T\}\_1]\backslash !],~[\backslash ![x\_2]\backslash !]\; \backslash in\; [\backslash ![\backslash mathsf\{T\}\_2]\backslash !],~\backslash ldots,~[\backslash ![x\_n]\backslash !]\; \backslash in\; [\backslash ![\backslash mathsf\{T\}\_n]\backslash !]$.

The {{math|n}}-tuple of type theory has the natural interpretation as an {{math|n}}-tuple of set theory:Steve Awodey, [http://www.andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf From sets, to types, to categories, to sets], 2009, preprint

: $[\backslash ![(x\_1,\; x\_2,\; \backslash ldots,\; x\_n)]\backslash !]\; =\; (\backslash ,[\backslash ![x\_1]\backslash !],\; [\backslash ![x\_2]\backslash !],\; \backslash ldots,\; [\backslash ![x\_n]\backslash !]\backslash ,)$

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

• Arity
• Coordinate vector
• Exponential object
• Formal language
• Multidimensional Expressions (OLAP)
• Prime k-tuple
• Relation (mathematics)
• Sequence
• Tuplespace
• Tuple Names

{{notelist}}

{{Reflist}}

• {{citation|first1=John P.|last1=D'Angelo|first2=Douglas B.|last2=West|title=Mathematical Thinking/Problem-Solving and Proofs|year=2000|edition=2nd|publisher=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, [https://books.google.com/books?id=ah2bwOwc06MC 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, [https://books.google.com/books?as_isbn=9780521753746 Lecture Notes in Logic and Set Theory. Volume 2: Set Theory], Cambridge University Press, 2003, {{isbn|978-0-521-75374-6}}, pp. 182–193

• {{Wiktionary-inline|tuple}}

{{Set theory}}

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