January 9th, 2014, 02:43 AM  #1 
Newbie Joined: Jan 2014 Posts: 23 Thanks: 0  Axiom of Choice
I do some exercises in set theory. Among them, there is one that says: Prove, using the Axiom of Choice, the following: a) b) It is clear to me that point a) requires AC (to "define" the function f(x)). However, I am not sure why is it required for b)? I solve b) as: My questions are: 1) Is my solution correct? 2) Does my solution require the AC? In some implicit way maybe? Maybe it is required for proving the implication ? 3) If not, why use AC? Maybe there is a more elegant way to solve b) using AC (or point a) for that matter, which requires AC)? Note 1: this is not a homework so don't tell me to go and ask the lecturer. I don't have one, that's why I am asking the question here. Note 2: my question concerns only point b). Point a) is clear enough. 
January 9th, 2014, 01:12 PM  #2  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Axiom of Choice Quote:
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January 10th, 2014, 08:40 AM  #3  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Re: Axiom of Choice Quote:
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January 11th, 2014, 01:27 AM  #4 
Newbie Joined: Jan 2014 Posts: 23 Thanks: 0  Re: Axiom of Choice
Evgeny, thank you for the answers. Yes, is any family of sets labeled by two indices. Good idea to use righttoleft a) to prove b). Maybe this is what was meant by the author of the book? I agree with you, however, that AC is not required in this case also. Thanks for your help, and best regards 

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