# Tuple

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 monoid (free semi-group with identity) generated by $X$; a function defined on the first $n$ natural numbers ($n\geq0$) with values in $X$.

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 $(x_1,\ldots,x_{n+1})=\{(x_1,\ldots,x_n),\{x_{n+1}\}\}$.