Can anyone help me with this please?
Consider the finite set $ S = \{a, b, c, d\}$
Suppose we know the function $f : S \rightarrow S $ has the property that
$f(a) = b, f(c) = d, f(b) $is not equal to $ b\ , f(d)$ is not equal to $ d$.
Prove that the dynamical system $(S, f)$ does not have a fixed point.
$f(a)=b \neq a$ so $a$ is not fixed.
$f(b) \neq b$ so $b$ is not fixed.
$f(c)=d \neq c$ so $c$ is not fixed.
$f(d)\neq d$ so $d$ is not fixed.
As all four points of $S$ are not fixed by $f$, the dynamical system $(S,f)$ has no fixed points.