Chapter 1.2, Problem 27E

Let $f:A\to B$, where $A$ and $B$ are nonempty. Prove that $f$ has the property that $f^{-1}\left(f\left(S\right)\right)=S$ for every subset $S$ of $A$ if and only if $f$ is one-to-one. (Compare with Exercise 15 b.).

b. For the mapping $f$, show that if $S=\left\{1,2\right\}$, then $f^{-1}\left(f\left(S\right)\right)\ne S$.