Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$ be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that $\frac{A}{S}=g(\frac{B}{f(S)})$.
Problem
Source: 17-th Iranian Mathematical Olympiad 1999/2000
Tags: function, algebra proposed, algebra