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.
Ordered Quadruple
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:
- $\tuple a := a$