Friday, 19 September 2014

elementary set theory - Sur- in- bijections and cardinality.




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:AB and g:BA, then |A|=|B| (|A||B| and |B||A|).



If there exists a surjection f:AB and an injection g:AB, 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:AB
implies an injection e:BA: for bB choose e(b)f1(b).
Together with the what you know about injections, this gives you everything you want.


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