for functions $f: A \to B$,$g:P(A) \to P(B)$ when $g(Y) = f^{-1}[Y]$, prove that $f$ is surjective iff $g$ is injective
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.