Definition:Ordered Tuple

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

If $\left \langle {a_k} \right \rangle_{k \in \N^*_n}$ 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.

An alternative notation is $\langle{a_1, a_2, \ldots, a_n}\rangle$

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 $\left \langle {a_k} \right \rangle_{1 \le k \le n}$.

It will be understood that the subscript runs through all values of $\N^*_n$ from $1$ to $n$.

Ordered n-tuple Defined by a Sequence
Let $\left \langle {a_k} \right \rangle_{k \in A}$ be a finite sequence of $n$ terms.

Let $\sigma$ be a permutation of $A$.

Then the ordered $n$-tuple defined by the sequence $\left \langle {a_k} \right \rangle_{k \in A}$ is the ordered $n$-tuple:
 * $\left \langle {a_{\sigma \left({\tau \left({j}\right)}\right)}}\right \rangle$

where $\tau$ is the unique isomorphism from the totally ordered set $\left[{1 .. n}\right]$ onto the totally ordered set $A$.

Equality of Ordered n-Tuples
Let:
 * $(1): \quad \left \langle {a_m} \right \rangle = \left({a_1, a_2, \ldots, a_m}\right)$

and
 * $(2): \quad \left \langle {b_n} \right \rangle = \left({b_1, b_2, \ldots, b_n}\right)$

be ordered tuples for some $m, n \in \N^*$. Then:


 * $\left \langle {a_m} \right \rangle = \left \langle {b_n} \right \rangle \iff n = m \land \forall j \in \N^*_n: a_j = b_j$

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

Notice the difference between ordered pairs and ordered couples.

By definition, an ordered couple $\left({a, b}\right)$ is in fact the set $\left\{{\left({1, a}\right), \left({2, b}\right)}\right\}$, where each of $\left({1, a}\right)$ and $\left({2, b}\right)$ are ordered pairs.

It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple.

However, in view of the equality of ordered tuples, it is generally accepted that it is valid to use the notation $\left({a, b}\right)$ to mean both an ordered couple and an ordered pair.

It is worth bearing this in mind, as there are times when it is important not to confuse them.

When such a distinction is important, the alternative notation $\langle{a,b}\rangle$ to mean $f\left({1}\right) = a$, $f\left({2}\right) = b$ is recommended.

Also see

 * Ordered Tuple as Ordered Set