Fixed Point of Idempotent Mapping

Theorem

Let $S$ be a set.

Let $f: S \to S$ be an idempotent mapping.

Let $f \left[{S}\right]$ be the image of $S$ under $f$.

Let $x \in S$.

Then $x$ is a fixed point of $f$ if and only if $x \in f \left[{S}\right]$.

Proof

Necessary Condition

Let $x$ be a fixed point of $f$.

Then:

$f \left({x}\right) = x$

and so by the definition of image of mapping:

$x \in f \left[{S}\right]$

$\Box$

Sufficient Condition

Let $x \in f \left[{S}\right]$.

Then by the definition of image:

$\exists y \in S: f \left({y}\right) = x$

Then:

 $\displaystyle x$ $=$ $\displaystyle f \left({y}\right)$ $\displaystyle \implies \ \$ $\displaystyle f \left({x}\right)$ $=$ $\displaystyle f \left({f \left({y}\right)}\right)$ Definition of Mapping $\displaystyle$ $=$ $\displaystyle f \left({y}\right)$ Definition of Idempotent Mapping $\displaystyle$ $=$ $\displaystyle x$

Thus by definition $x$ is a fixed point of $f$.

$\blacksquare$