Definition:Ordered Tuple

An ordered $$n$$-tuple is a sequence whose domain is $$\mathbb{N}^*_n$$.

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

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 $$\left \langle {a_k} \right \rangle_{1 \le k \le n}$$. It will be understood that the subscript runs through all values of $$\mathbb{N}^*_n$$ from $$1$$ to $$n$$.

Ordered n-tuple Defined by a Sequence
Let $$\left \langle {a_k} \right \rangle_{k \in A}$$ be a 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) $$\left \langle {a_m} \right \rangle = \left({a_1, a_2, \ldots, a_m}\right)$$, and
 * 2) $$\left \langle {b_n} \right \rangle = \left({b_1, b_2, \ldots, b_n}\right)$$

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

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

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".

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.