(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove, without using the Axiom of Choice:

if f: X->Y is surjective and Y is finite, there exists a 'section', a function s:Y->X such that f(s(y))=y for all y in Y

Hint: perform induction over the cardinality of Y

3. The attempt at a solution

Induction over the cardinality of Y? I don't even know what they mean. I'd say this would require the AC because we need to pick one of the elements of f^-1 (y) for all y in Y. since X is infinite, one of these f^-1 (y) has to be infinite as well, and I don't know how to pick a specific element of this if I can't just say 'I pick some'.

And I also don't understand the hint. Some help would be appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Finite Axiom of Choice

**Physics Forums | Science Articles, Homework Help, Discussion**