Equivalent statement of AC or not?

Asked Dec 14 '20 at 13:34

Active Dec 14 '20 at 13:45

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Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not AC can be derived from this proposition, in return? I have tried but haven't found a proof.

Any help would be appreciated.

asked Dec 14 '20 at 13:34

Jihai Zhu

see this answer https://math.stackexchange.com/questions/108810/does-a-injective-function-f-a-to-b-and-surjective-function-g-a-to-b-impl – Evangelopoulos Foivos Dec 14 '20 at 13:39
I wrote about this extensively. – Asaf Karagila Dec 14 '20 at 13:43
http://karagila.org/2014/on-the-partition-principle/#:~:text=The%20Partition%20Principle%20%28PP%29%20is%20the%20principle%20stating,B%20parts%2C%20we%20could%20only%20decrease%20the%20cardinality. – Jihai Zhu Dec 14 '20 at 14:30
I think the answer can be found here that the statement might be a weakening of AC. – Jihai Zhu Dec 14 '20 at 14:32
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The answer is that it is an open problem. – Asaf Karagila Dec 14 '20 at 14:49

Equivalent statement of AC or not?

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