Fixed Point of Idempotent Mapping

Theorem

Let $S$ be a set.

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

Let $\Img f$ be the image of $f$.

Let $x \in S$.

Then $x$ is a fixed point of $f$ if and only if $x \in \Img f$.

Proof

Necessary Condition

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

Then:

$\map f x = x$

and so by definition of image of mapping:

$x \in \Img f$

$\Box$

Sufficient Condition

Let $x \in \Img f$.

Then by the definition of image:

$\exists y \in S: \map f y = x$

Then:

\(\ds x\)

\(=\)

\(\ds \map f y\)

\(\ds \leadsto \ \ \)

\(\ds \map f x\)

\(=\)

\(\ds \map f {\map f y}\)

Definition of Mapping

\(\ds \)

\(=\)

\(\ds \map f y\)

Definition of Idempotent Mapping

\(\ds \)

\(=\)

\(\ds x\)

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

$\blacksquare$