Definition:Power of Mapping

Definition
Let $$f: S \to S$$ be a mapping from $$S$$ to itself.

Because the domain of $$f$$ is equal to the codomain of $$f$$ (both are $$S$$), the composite mapping $$f \circ f$$ is defined.

We define the $$n$$th power of $$f$$ as:


 * $$\forall n \in \N: f^n = \begin{cases}

I_S & : n = 0 \\ f \circ f^{n-1} & : n > 0 \end{cases}$$ where $$I_S$$ is the identity mapping.