Definition:Ordered Tuple

Definition
Let $n \in \N_{>0}$.

Let $\N^*_n$ be the first $n$ non-zero natural numbers:
 * $\N^*_n := \left\{{1, 2, \ldots, n}\right\}$

An ordered tuple (of length $n$) is a finite sequence whose domain is $\N^*_n$.

Also defined as
Some treatments take the intuitive approach of regarding an ordered tuple merely as an ordered set, that is, without stressing the fact of it being a mapping from a subset of the natural numbers.

Also known as
The term ordered $n$-tuple can sometimes be seen, particularly for specific instances of $n$.

Instead of writing 2-tuple, 3-tuple and 4-tuple, the terms couple, triple and quadruple are usually used.

Some sources refer to an ordered tuple as a tuple.

Also denoted as
Various alternatives to $\left({a_1, a_2, \ldots, a_n}\right)$ can be found in the literature, for example:
 * $\left \langle {a_1, a_2, \ldots, a_n} \right \rangle$

This notation is recommended when use of parentheses would be ambiguous.

There are also specialised instances of an ordered tuple where the convention is to use angle brackets.

Other notations which may be encountered are:
 * $\left[{a_1, a_2, \ldots, a_n}\right]$
 * $\left\{{a_1, a_2, \ldots, a_n}\right\}$

but both of these are strongly discouraged: the square bracket format because there are rendering problems on this site, the latter because it is too easily confused with set notation.

An ordered tuple can (and often will) be denoted $\left({a_1, a_2, \ldots, a_n}\right)$ instead of by $\left \langle {a_k} \right \rangle_{1 \le k \le n}$ etc.

As an example, $\left({6, 3, 3}\right)$ is the ordered triple $f$ defined as:
 * $f \left({1}\right) = 6, f \left({2}\right) = 3, f \left({3}\right) = 3$

In order to further streamline notation, it is common to use the more compact $\left \langle {a_n} \right \rangle$ for $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$.

Also see

 * Definition:Ordered Tuple as Ordered Set


 * Definition:Finite Sequence: Note that an ordered tuple and a finite sequence are in fact the same thing. However, with an ordered tuple the emphasis is usually placed on the image set, while for a finite sequence the domain is usually more conceptually important, and can in fact be considered as any finite set.