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Consider the following statement: for any set $E$ and $G\subseteq E\times E$, you can to get a function $f:A\rightarrow B$ where $A=dom G$, $B=ran G$ and $f\subseteq G$.

I want to show that this implies axiom of choice.

My attempt:I will prove that the statement above implies the existence of a choice function. Indeed, let A an arbitrary set and consider $E=\mathcal{P}(A)\setminus\{\emptyset\}$ and take $G:=\{(B,\{x\})\subseteq E\times E:x\in B\}$. So, by hypotesis exists a function $f:E\rightarrow A$ that it is, in fact, a choice function.

My doubts in the proof is with the set $G$, I don't know if in your definition I use axiom of choice, because it seems that I do infinitely many choices.

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  • $\begingroup$ Did you intend to require that (the graph of) $f$ is a subset of $G$? Without such a requirement, your statement does not imply the axiom of choice; in fact it's provable from the other axioms of set theory. The reason is that there are functions from any set $A$ to any set $B$ unless $A\neq\varnothing=B$. And that exceptional situation can't occur for you because, if the range of $G$ is empty, then so is $G$ and therefore so is the domain of $G$. $\endgroup$ – Andreas Blass Oct 27 '18 at 20:16
  • $\begingroup$ @AndreasBlass Yes, I did forget it hypothesis. It is fixed $\endgroup$ – Gödel Oct 27 '18 at 20:58
  • $\begingroup$ Could you mention the source of the statement that you are proving equivalent to choice? @Gödel $\endgroup$ – Carl Mummert Oct 28 '18 at 1:29
  • $\begingroup$ @Carl: It's a fairly well-known version. I couldn't even tell you who proved that equivalence. $\endgroup$ – Asaf Karagila Oct 28 '18 at 8:51
  • $\begingroup$ @Asaf: it's pretty clearly equivalent to AC, too :). The OP must have found it somewhere - that source might be interesting for others. $\endgroup$ – Carl Mummert Oct 28 '18 at 14:19
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Yes, you have a problem there. Specifically, your function returns singletons of elements of $A$, not elements of $A$.

The obvious way to correct this is to take $E=A\cup\mathcal P(A)$ and $G=\{(B,x)\mid x\in B\subseteq A\}$.

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  • $\begingroup$ It makes sense to me. $\endgroup$ – Gödel Oct 27 '18 at 17:34
  • $\begingroup$ Is it not necessary delet $\emptyset$ from $E$? $\endgroup$ – Gödel Oct 27 '18 at 17:38
  • $\begingroup$ No, because it is not in the domain of $G$. $\endgroup$ – Asaf Karagila Oct 27 '18 at 18:31
  • $\begingroup$ Of course I see it $\endgroup$ – Gödel Oct 27 '18 at 18:44

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