Chapter 1.2, Problem 28E

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

Exercise 15c.

c. For this same $f$ and $T=\left\{4,9\right\},$ show that $f\left(f^{-1}\left(T\right)\right)\ne T$.