Saturday, December 17, 2016

elementary set theory - Proving (AleB)vee(BleA) for sets A and B

For any pair of sets A and B, we can define AB iff there exists injection f:AB. I am trying prove that (AB)(BA).


I have tried assuming ¬(AB), then proving BA by constructing the required injection, but I haven't been able to make any progress. Any hints, etc. would be appreciated.


EDIT


Assuming ¬(AB), can you prove there exists a surjection f:AB? Then it would be easy, by applying AC, to construct an injection g:BA

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