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A finite sequence (admitting repetitions) of elements from some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944301.png" />. A tuple is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944307.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944308.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t0944309.png" /> is called its length (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443010.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443011.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443012.png" />-th term of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443013.png" />-tuple and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443015.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443016.png" /> one finds the empty tuple, which contains no terms.
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A finite sequence (admitting repetitions) of elements from some set $X$. A tuple is denoted by $\langle x_1,\ldots,x_n\rangle$, $(x_i)$, $(x_i)_{i=1}^n$, $(x_i)_1^n$, $(x_i)_{i\in\{1,\ldots,n\}}$, $(x_1,\ldots,x_n)$, or $x_1,\ldots,x_n$. The number $n$ is called its length ($n\geq0$), $x_i$ is called the $i$-th term of the $n$-tuple and $x_i\in X$ ($1\leq i\leq n$). For $n=0$ one finds the empty tuple, which contains no terms.
  
Synonyms of the term tuple are the following: a [[Word|word]] in the [[Alphabet|alphabet]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443017.png" /> (in this case it is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443018.png" /> is finite); an element of some Cartesian power of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443019.png" />; an element of the free semi-group with identity generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443020.png" />; a function defined on the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443021.png" /> natural numbers (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443022.png" />) with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443023.png" />.
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Synonyms of the term tuple are the following: a [[Word|word]] in the [[Alphabet|alphabet]] $X$ (in this case it is usually assumed that $X$ is finite); an element of some Cartesian power of the set $X$; an element of the free semi-group with identity generated by $X$; a function defined on the first $n$ natural numbers ($n\geq0$) with values in $X$.
  
  
  
 
====Comments====
 
====Comments====
The typical property of tuples is that a tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443024.png" /> is equal to another one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443025.png" /> precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443028.png" />. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel [[Set theory|set theory]]), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443029.png" /> is the empty set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443030.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094430/t09443031.png" />.
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The typical property of tuples is that a tuple $(x_1,\ldots,x_n)$ is equal to another one $(y_1,\ldots,y_m)$ precisely when $n=m$ and $x_i=y_i$ for all $i=1,\ldots,n$. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel [[Set theory|set theory]]), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure: $(x_1,\ldots,x_n)$ is the empty set for $n=0$, and $(x_1,\ldots,x_{n+1})=\{(x_1,\ldots,x_n),\{x_{n+1}\}\}$.

Revision as of 10:27, 17 April 2014

A finite sequence (admitting repetitions) of elements from some set $X$. A tuple is denoted by $\langle x_1,\ldots,x_n\rangle$, $(x_i)$, $(x_i)_{i=1}^n$, $(x_i)_1^n$, $(x_i)_{i\in\{1,\ldots,n\}}$, $(x_1,\ldots,x_n)$, or $x_1,\ldots,x_n$. The number $n$ is called its length ($n\geq0$), $x_i$ is called the $i$-th term of the $n$-tuple and $x_i\in X$ ($1\leq i\leq n$). For $n=0$ one finds the empty tuple, which contains no terms.

Synonyms of the term tuple are the following: a word in the alphabet $X$ (in this case it is usually assumed that $X$ is finite); an element of some Cartesian power of the set $X$; an element of the free semi-group with identity generated by $X$; a function defined on the first $n$ natural numbers ($n\geq0$) with values in $X$.


Comments

The typical property of tuples is that a tuple $(x_1,\ldots,x_n)$ is equal to another one $(y_1,\ldots,y_m)$ precisely when $n=m$ and $x_i=y_i$ for all $i=1,\ldots,n$. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel set theory), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure: $(x_1,\ldots,x_n)$ is the empty set for $n=0$, and $(x_1,\ldots,x_{n+1})=\{(x_1,\ldots,x_n),\{x_{n+1}\}\}$.

How to Cite This Entry:
Tuple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tuple&oldid=31808
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article