In mathematics, a tuple is a finite sequence or ordered list of or, more generally, mathematical objects, which are called the elements of the tuple. An tuple is a tuple of elements, where is a nonnegative integer. There is only one 0tuple, called the empty tuple. A 1tuple and a 2tuple are commonly called a singleton and an ordered pair, respectively.
Tuple may be formally defined from ordered pairs by recurrence by starting from ; indeed, a tuple can be identified with the ordered pair of its first elements and its th element.
Tuples are usually written by listing the elements within parentheses "", separated by a comma and a space; for example, denotes a 5tuple. 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 , 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.
[
]
Etymology
The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., ‑tuple, ..., where the prefixes are taken from the
Latin names of the numerals. The unique 0tuple 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 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" (threefold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek language ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".[ OED, s.v. "triple", "quadruple", "quintuple", "decuple"]
Properties
The general rule for the identity of two 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:

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

Tuple elements are ordered: tuple $(1,2,3)\; \backslash neq\; (3,2,1)$, but set $\backslash \{1,2,3\backslash \}\; =\; \backslash \{3,2,1\backslash \}$.

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
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:
 $F\; ~:=~\; \backslash left\backslash \{\; \backslash left(1,\; a\_1\backslash right),\; \backslash ldots,\; \backslash left(n,\; a\_n\backslash right)\; \backslash right\backslash \}.$
Tuples as nested ordered pairs
Another way of modeling tuples in Set Theory is as nested
. This approach assumes that the notion of ordered pair has already been defined.

The 0tuple (i.e. the empty tuple) is represented by the empty set $\backslash emptyset$.

An tuple, with , can be defined as an ordered pair of its first entry and an tuple (which contains the remaining entries when :

: $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; (a\_1,\; (a\_2,\; a\_3,\; \backslash ldots,\; a\_n))$
This definition can be applied recursively to the 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:

The 0tuple is the empty set $\backslash emptyset$.

For :

: $(a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_n)\; =\; ((a\_1,\; a\_2,\; a\_3,\; \backslash ldots,\; a\_\{n1\}),\; a\_n)$
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}
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 0tuple (i.e. the empty tuple) is represented by the empty set $\backslash emptyset$;

Let $x$ be an 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}
tuples of sets
In discrete mathematics, especially
combinatorics and finite probability theory, tuples arise in the context of various counting problems and are treated more informally as ordered lists of length .
tuples whose entries come from a set of elements are also called
arrangements with repetition,
permutations of a multiset and, in some nonEnglish literature,
variations with repetition. The number of tuples of an set is . This follows from the combinatorial rule of product.
If is a finite set of
cardinality , this number is the cardinality of the fold Cartesian power . 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:
 $(x\_1,\; x\_2,\; \backslash ldots,\; x\_n)\; :\; \backslash mathsf\{T\}\_1\; \backslash times\; \backslash mathsf\{T\}\_2\; \backslash times\; \backslash ldots\; \backslash times\; \backslash mathsf\{T\}\_n$
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.
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 theory of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets $S\_1,\; S\_2,\; \backslash ldots,\; S\_n$ (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 tuple of type theory has the natural interpretation as an tuple of set theory:[Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint]
 $\backslash ![(x\_1,\backslash !]\; =\; (\backslash ,\backslash ![x\_1\backslash !],\; \backslash ![x\_2\backslash !],\; \backslash ldots,\; \backslash ![x\_n\backslash !]\backslash ,)$
The
unit type has as semantic interpretation the 0tuple.
See also
Notes
Sources

Keith Devlin, The Joy of Sets. Springer Verlag, 2nd ed., 1993, , pp. 7–8

Abraham Adolf Fraenkel, Yehoshua BarHillel, Azriel Lévy, Foundations of school Set Theory, Elsevier Studies in Logic Vol. 67, 2nd Edition, revised, 1973, , p. 33

Gaisi Takeuti, W. M. Zaring, Introduction to Axiomatic Set Theory, Springer GTM 1, 1971, , p. 14

George J. Tourlakis, Lecture Notes in Logic and Set Theory. Volume 2: Set Theory, Cambridge University Press, 2003, , pp. 182–193
External links