# Tuple

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A tuple is an ordered list of elements. In mathematics, an n-tuple is a sequence (or ordered list) of ${\displaystyle n}$ elements, where ${\displaystyle n}$ is a non-negative integer. There is only one 0-tuple, an empty sequence. An ${\displaystyle n}$-tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "${\displaystyle ({\text{ }})}$" and separated by commas; for example, ${\displaystyle (2,7,4,1,7)}$ denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets "${\displaystyle [{\text{ }}]}$" or angle brackets "${\displaystyle \langle {\text{ }}\rangle }$". Braces "${\displaystyle \{\}}$" are almost never used for tuples, as they are the standard notation for sets. Tuples are often used to describe other mathematical objects, such as vectors. In computer science, tuples are directly implemented as product types in most functional programming languages. More commonly, they are implemented as record types, where the components are labeled instead of being identified by position alone. This approach is also used in relational algebra. Tuples are also used in relation to programming the semantic web with Resource Description Framework or RDF. Tuples are also used in linguistics[1] and philosophy.[2]

## Etymology

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. n 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 a medieval Latin suffix ‑plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".[3]

### Names for tuples of specific lengths

Tuple Length ${\displaystyle n}$ Name Alternative names
0 empty tuple unit
1 single singleton
2 double couple / pair / dual / twin
3 triple treble / triplet
5 quintuple
6 sextuple hextuple
7 septuple
8 octuple
9 nonuple
10 decuple
11 undecuple hendecuple
12 duodecuple
13 tredecuple
100 centuple

## Properties

The general rule for the identity of two ${\displaystyle n}$-tuples is

${\displaystyle (a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})}$ if and only if ${\displaystyle a_{1}=b_{1},{\text{ }}a_{2}=b_{2},{\text{ }}\ldots ,{\text{ }}a_{n}=b_{n}.}$

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

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

## Definitions

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

### Tuples as functions

If we are dealing with sets, an ${\displaystyle n}$-tuple can be regarded as a function, F, whose domain is the tuple's implicit set of element indices, X, and whose codomain, Y, is the tuple's set of elements. Formally:

${\displaystyle (a_{1},a_{2},\dots ,a_{n})\equiv (X,Y,F)}$

where:

{\displaystyle {\begin{aligned}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{aligned}}}

In slightly less formal notation this says:

${\displaystyle (a_{1},a_{2},\dots ,a_{n}):=(F(1),F(2),\dots ,F(n)).}$

### 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; thus a 2-tuple

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set ${\displaystyle \emptyset }$.
2. An ${\displaystyle n}$-tuple, with ${\displaystyle n>0}$, can be defined as an ordered pair of its first entry and an ${\displaystyle (n-1)}$-tuple (which contains the remaining entries when ${\displaystyle n>1}$):
${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))}$

This definition can be applied recursively to the ${\displaystyle (n-1)}$-tuple:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))}$

Thus, for example:

{\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:

This definition can be applied recursively:

${\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})}$

Thus, for example:

{\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, 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 ${\displaystyle \emptyset }$;
2. Let ${\displaystyle x}$ be an ${\displaystyle n}$-tuple ${\displaystyle (a_{1},a_{2},\ldots ,a_{n})}$, and let ${\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)}$. Then, ${\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}}$. (The right arrow, ${\displaystyle \rightarrow }$, could be read as "adjoined with".)

In this formulation:

${\displaystyle player\leq score}$