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In mathematics a tuple is a finite ordered list (sequence) of elements. An Template:Math-tuple is a sequence (or ordered list) of Template:Math elements, where Template:Math is a non-negative integer. There is only one 0-tuple, an empty sequence. An Template:Math-tuple is defined inductively using the construction of an ordered pair.

Mathematicians usually write tuples by listing the elements within parentheses "$(\text{ })$" 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 only used in defining arrays in some programming languages such as Java and Visual Basic, 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. In dynamically typed languages, such as Lisp, lists are commonly used as tuples.Template:Citation needed Most typed functional programming|functional programming languages implement tuples directly as product types,[1] tightly associated with Algebraic data type|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]

## EtymologyEdit

The term originated as an abstraction of the sequence: single, 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. A 1‑tuple is called a singleton, a 2‑tuple is called an ordered pair and a 3‑tuple is a triple or triplet. Template:Math can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple, 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".[6]Template:Efn

## PropertiesEdit

The general rule for the identity of two Template:Math-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.

1. A tuple may contain multiple instances of the same element, so
tuple $(1,2,2,3) \neq (1,2,3)$; but set $\{1,2,2,3\} = \{1,2,3\}$.
2. Tuple elements are ordered: tuple $(1,2,3) \neq (3,2,1)$, but set $\{1,2,3\} = \{3,2,1\}$.
3. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.

## DefinitionsEdit

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

### Tuples as functionsEdit

If we are dealing with sets, an Template:Math-tuple can be regarded as a function, Template:Math, whose domain is the tuple's implicit set of element indices, Template:Math, and whose codomain, Template:Math, is the tuple's set of elements. Formally:

$(a_1, a_2, \dots, a_n) \equiv (X,Y,F)$

where:

\begin{align} X & = \{1, 2, \dots, n\} \\ Y & = \{a_1, a_2, \ldots, a_n\} \\ F & = \{(1, a_1), (2, a_2), \ldots, (n, a_n)\}. \\ \end{align}

In slightly less formal notation this says:

$(a_1, a_2, \dots, a_n) := (F(1), F(2), \dots, F(n)).$

### Tuples as nested ordered pairsEdit

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; thus a 2-tuple

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$.
2. An Template:Math-tuple, with Template:Math, can be defined as an ordered pair of its first entry and an Template:Math-tuple (which contains the remaining entries when Template:Math:
$(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, a_3, \ldots, a_n))$

This definition can be applied recursively to the Template:Math-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{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 $\emptyset$.
2. For Template:Math:
$(a_1, a_2, a_3, \ldots, a_n) = ((a_1, a_2, a_3, \ldots, a_{n-1}), a_n)$

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 Edit

 Jake

### Tuples as nested setsEdit

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 $\emptyset$;
1. Voting
Xander
##### Edit

. (The right arrow, $\rightarrow$, could be read as "adjoined with".)

In this formulation:

## n-tuples of m-sets Edit

In discrete mathematics, especially combinatorics and finite probability theory, Template:Math-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length Template:Math.[7] Template:Math-tuples whose entries come from a set of Template:Math elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations]] with repetition. The number of Template:Math-tuples of an Template:Math-set is Template:Math. This follows from the combinatorial rule of product.[8] If Template:Math is a finite set of cardinality Template:Math, this number is the cardinality of the Template:Math-fold Cartesian power Template:Math. Tuples are elements of this product set.

## Type theory Edit

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:

$(x_1, x_2, \ldots, x_n) : \mathsf{T}_1 \times \mathsf{T}_2 \times \ldots \times \mathsf{T}_n$

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 Template:Math-tuple of type theory has the natural interpretation as an Template:Math-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.