# Definition:Ordered Tuple as Ordered Set

## Definition

The rigorous definition of an ordered tuple is as a finite 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.

### Ordered Triple

The ordered triple $\tuple {a, b, c}$ of elements $a$, $b$ and $c$ can be defined either as the ordered pair:

$\tuple {a, \tuple {b, c} }$

or as the ordered pair:

$\tuple {\tuple {a, b}, c}$

where $\tuple {a, b}$ and $\tuple {b, c}$ are themselves ordered pairs.

The ordered quadruple $\tuple {a, b, c, d}$ of elements $a$, $b$, $c$ and $d$ is defined either as the ordered pair:

$\tuple {a, \tuple {b, c, d} }$

or:

$\tuple {\tuple {a, b, c}, d}$

where $\tuple {a, b, c}$ and $\tuple {b, c, d}$ are themselves ordered triples.

### Ordered Tuple

The ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ of elements $a_1, a_2, \ldots, a_n$ is defined as either the ordered pair:

$\tuple {a_1, \tuple {a_2, a_3, \ldots, a_n} }$

or as the ordered pair:

$\tuple {\tuple {a_1, a_2, \ldots, a_{n - 1} }, a_n}$

where $\tuple {a_2, a_3, \ldots, a_n}$ and $\tuple {a_1, a_2, \ldots, a_{n - 1} }$ are themselves ordered tuples.

### Ordered Singleton

The degenerate case of the ordered tuple $\tuple a$ on the singleton $\set a$ is defined as:

$\tuple a := a$