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 $n$-tuple is a finite sequence whose domain is $\N^*_n$.

Term of Ordered Tuple
If $\left \langle {a_k} \right \rangle_{k \mathop \in \N^*_n}$ is an ordered $n$-tuple, then $a_k$ is called the $k$th term of the ordered $n$-tuple for each $k \in \N^*_n$.

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

An ordered $n$-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 by $f \left({1}\right) = 6, f \left({2}\right) = 3, f \left({3}\right) = 3$.

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

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

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.

Also see

 * Definition:Ordered Tuple as Ordered Set