# Definition:Recursively Defined Mapping/Natural Numbers

## Definition

Let $p \in \N$ be a natural number.

Let $S = \set {x \in \N: p \le x}$.

Let $T$ be a set.

Let $g: T \to T$ be a mapping.

Let $f: S \to T$ be the mapping defined as:

$\forall x \in S: \map f x = \begin{cases} a & : x = p \\ \map g {\map f n} & : x = n + 1 \end{cases}$

for $a \in T$.

Then $f$ is said to be recursively defined on $S$.