Question 30
C7 (IRE) Let $U$ be a finite set and let $f,g$ be bijective functions from $U$ onto itself. Let
$$S=\{w\in U:f(f(w))=g(g(w))\},\qquad T=\{w\in U:f(g(w))=g(f(w))\},$$
and suppose that $U=S\cup T$. Prove that for $w\in U$, $f(w)\in S$ if and only if $g(w)\in S$.
$$S=\{w\in U:f(f(w))=g(g(w))\},\qquad T=\{w\in U:f(g(w))=g(f(w))\},$$
and suppose that $U=S\cup T$. Prove that for $w\in U$, $f(w)\in S$ if and only if $g(w)\in S$.