Re: axiom of choice and equivalent statements
- From: "Norm Dresner" <ndrez@xxxxxxx>
- Date: Sun, 05 Feb 2006 15:58:53 GMT
"Li Yi" <liyi.cn@xxxxxxxxx> wrote in message news:1139145297.879035.68570@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Prove the following statements are equivalent:
(1) axiom of choice;
(2) for any sets A and B, it holds either A <= B or B <= A. (X <= Y
means that there is an injection from X to Y)
(3) Zorn's lemma
(4) Well-ordering Principle : every set can be well-ordered.
(5) Every binary relation can be single-valued: Given a binary relation
R, there exists a function F subset R such that dom F = dom R
(6) A is a nonempty set, then A^a is nonempty for any ordinal number a.
I know that (1)(3)(4) are equivalent and have a proof of (3)=>(2).
The simplest solution to your problem is to get from your school library a copy of "Equivalents of the Axiom of Choice". First edition is Rubin & Rubin. Second addition added someone else whose name I can't recall.
Norm
.
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