I think about surjection, injection and bijections from A to B as ≥, ≤, and = respectively in terms of cardinality. Is this correct? And extrapolating from that, are these theorems correct?
If there exist two surjections f:A→B and g:B→A, then |A|=|B| (|A|≥|B| and |B|≥|A|).
If there exists a surjection f:A→B and an injection g:A→B, then |A|=|B| (|A|≥|B| and |A|≤|B|).
Are these theorems correct?
I know the case for two injections is true.
Answer
If you assume the axiom of choice, then the existence of a surjection f:A→B
implies an injection e:B→A: for b∈B choose e(b)∈f−1(b).
Together with the what you know about injections, this gives you everything you want.
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