Definition:Ordered Tuple as Ordered Set

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Definition

The rigorous definition of an ordered tuple is as a sequence whose domain is $\N^*_n$.

However, it is possible to treat an ordered tuple as an extension of the concept of an ordered pair.

The ordered triple $\left({a, b, c}\right)$ of elements $a$, $b$ and $c$ is defined as the ordered pair:

$\left({a, \left({b, c}\right)}\right)$

where $\left({b, c}\right)$ is itself an ordered pair.

Similarly, the ordered quadruple $\left({a, b, c, d}\right)$ of elements $a$, $b$, $c$ and $d$ is defined as the ordered pair:

$\left({a, \left({b, c, d}\right)}\right)$

where $\left({b, c, d}\right)$ is itself an ordered triple.

Similarly, the ordered tuple $\left({a_1, a_2, \ldots, a_n}\right)$ of elements $a_1, a_2, \ldots, a_n$ is defined as the ordered pair:

$\left({a_1, \left({a_2, a_3, \ldots, a_n}\right)}\right)$

where $\left({a_2, a_3, \ldots, a_n}\right)$ is itself an ordered tuple.

Some sources define the ordered tuple $\left({a_1, a_2, \ldots, a_n}\right)$ of elements $a_1, a_2, \ldots, a_n$ as the ordered pair:

$\left({\left({a_1, a_2, \ldots, a_{n-1}}\right), a_n}\right)$

Whichever definition is chosen does not matter much, as long as it is understood which is used. And even then, the importance is limited.