Question 41
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$ and let $T = f(S)$ denote the image of $S$ under $f$. Suppose that for every $g$ in $\mathcal{A}$ with $g \neq f$, we have $f \circ g \circ f \neq g \circ f \circ g$.
Prove that $f(T) = T$.
Prove that $f(T) = T$.