If I there exists an injection $\phi: S_1 \to S_2$ and a surjection $\tau: S_1 \to S_2$, does there necessarily exist a bijection between sets $S_1$ and $S_2$?
I'd like this to be true, but I don't see a way to construct a bijection directly from $\phi$ and $\tau$.
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$\begingroup$The statement that if there is a surjection from $A$ to $B$, then there is an injection from $B$ to $A$ is known as the Partition principle. It is a consequence of the axiom of choice, and it’s not known whether it is equivalent to the axiom of choice. Given the partition principle, the existence of both an injection and a surjection from $A$ to $B$ implies the existence of injections from $A$ to $B$ and $B$ to $A$, and the Schröder-Bernstein theorem then implies that there is a bijection from $A$ to $B$.
$\endgroup$ $\begingroup$Cantor-Bernstein-Schroeder Theorem
And the fact that, if there exists a surjection from A to B, then there exists an injection from B to A.
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