Sur- in- bijections and cardinality. I think about surjection, injection and bijections from $A$ to $B$ as $\ge$, $\le$, and $=$ respectively in terms of cardinality. Is this correct? And extrapolating from that, are these theorems correct? If there exist two surjections $f:A\rightarrow B$ and $g:B\rightarrow A$, then $|A|=|B|$ ($|A|\ge|B|$ and $|B|\ge|A|$). If there exists a surjection $f:A\rightarrow B$ and an injection $g:A\rightarrow B$, then $|A|=|B|$ ($|A|\ge|B|$ and $|A|\le|B|$). Are these theorems correct? I know the case for two injections is true.

If you assume the axiom of choice, then the existence of a surjection $f:A\to B$ implies an injection $e:B\to A$: for $b\in B$ choose $e(b)\in f^{-1}(b)$. Together with the what you know about injections, this gives you everything you want.