Exercise · ZW-AXIOM-OF-CHOICE-010 · ★★★ Advanced
A student claims that, without the axiom of choice , every surjection \(p:X\to Y\) has a right inverse . What is correct to say about this claim?
Choose one option.
A It always holds, because one can simply take \(p^{-1}\) as a function. B It is equivalent to the axiom of choice, so it cannot be proved in ZF in general. C It holds in ZF, because every surjection is injective. D It is false, because the right inverse coincides with \(p^{-1}\). Show answer B. It is equivalent to the axiom of choice, so it cannot be proved in ZF in general.← Back to the question
Explanation Every surjection has a right inverse if and only if the axiom of choice holds, so it is not provable in ZF in general. The other statements confuse surjectivity with injectivity or assume each preimage is a single element.
Statement that every family of nonempty sets admits a choice function \(f\) that picks one element from each set in the family.
\[\forall X\,(\emptyset\notin X \rightarrow \exists f\,\forall A\in X\,(f(A)\in A))\]
More practice problems: axiom of choice