Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.
I was wondering whether the following statements are true (maybe by using the AC if necessary):
- Suppose $f \colon A\to B$ and $g\colon B\to A$ are both surjective, does this imply that there is a bijection between $A$ and $B$.
- Suppose either $f\colon A\to B$ or $g\colon A\to B$ is surjective and the other one injective, does this imply that there is a bijection between $A$ and $B$.