Definition:Ordered Tuple

Definition
An ordered $$n$$-tuple is a 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.

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 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 $$m, n \in \N^*$$. Then:


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

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.

Also see

 * Ordered Tuple as Ordered Set