# Can a surjection and injection exist but not a bijection? [duplicate]

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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## marked as duplicate by Asaf Karagila, Chris Eagle, Jim, Andreas Caranti, Alexander Gruber♦Feb 26 '13 at 11:35

Depends of AC.. –  Gastón Burrull Feb 26 '13 at 1:19
@hchau, could you tell us the level that you're learning at? Depending on if you're learning elementary discrete maths, or axiomatic set theory, it can make a big difference to what form your answer takes. –  Dan Rust Feb 26 '13 at 1:23
Oh, so I haven't actually taken an undergraduate set theory course yet, but this question just popped into my mind, so I thought I'd ask it here. –  hchau Feb 26 '13 at 1:32
I have actually answered this question at least once on this site (and I believe that it was twice or thrice). –  Asaf Karagila Feb 26 '13 at 5:54
–  Asaf Karagila Feb 26 '13 at 6:06

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 $B$.