Axiom of choice, proving a function is onto

Question

I had some questions about the Axiom of choice.

suppose I have a function f:A->B, where A and B are infinite sets, and I have to prove f is onto.

So as a general strategy I pick an arbitrary element b in B, and I find an element a in A, such that f(a) =b.

My question is, is picking an arbitrary element in B making use of the axiom of choice?

Also... seems to me the axiom of choice would not be sufficient, because the way I understand it, the axiom of choice allows us to pick one element out of a set, but not necessarily every element.

But to prove B is onto, I'd have to be able to pick any and every element in B right?

AC allows you to pick any element. If you can pick any element and find its pre-image, then the function is "onto". — Terra Hyde, Sep 15 '15 at 14:36
You don't need AC to choose an element from a single set, or for that matter from any finite amount of sets. — vadim123, Sep 15 '15 at 14:36
How would you pick an element from an infinite set in general? — Ameet Sharma, Sep 15 '15 at 14:41
How do you pick an element from any set? What does the fact the set is infinite even tell you? — Asaf Karagila, Sep 15 '15 at 19:03

score 2 · Accepted Answer · answered Sep 15 '15 at 15:06

2

The axiom of choice is not needed. You are choosing one element $b$ from $B$, presumably a non-empty set. This uses one existential instantiation.

Then you prove the set of elements mapped to that chosen element is also non-empty, and you do it by showing there is some $a$ mapped to $b$.

You are not making infinitely many arbitrary choices, so the axiom of choice is not used. Not in "general" anyway.

answered Sep 15 '15 at 15:06

Asaf Karagila

405,794

Thanks. Still a little confused. Given an infinite set, is there a way to pick every element of the set? – Ameet Sharma Sep 15 '15 at 15:13
It's part of the inference rules of logic, existential instantantiation. – Asaf Karagila Sep 15 '15 at 15:31

Axiom of choice, proving a function is onto

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