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Math Help - Equivalence Class

  1. #1
    Super Member redsoxfan325's Avatar
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    Equivalence Class

    For x,y\in\mathbb{R}, say that x\sim y if |x-y|\in\mathbb{Q}. Define [x]=\{y: y\sim x\}. Let X=\{[x]: x\in\mathbb{R}\}.

    If I want (X,d) to be a metric space, is there a non-trivial metric d I can use that will make X complete? If not, what metric would make the most sense to use on this metric space?

    d([x],[y])=???

    Any suggestions?

    Note: I'm asking this out of intellectual curiosity. It is not a homework problem.
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  2. #2
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    I guess youv'e thought of it, but why not define it as:
    d([x],[y])=0 if [x]=[y] and |x-y| if |x-y| isn't rational (which means when [x] is different than [y]).

    This seems as a plausible metric for X.
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by InvisibleMan View Post
    why not define it as:
    d([x],[y])=0 if [x]=[y] and |x-y| if |x-y| isn't rational (which means when [x] is different than [y]).
    This metric is not well defined since we have [\sqrt{2}]=[\sqrt{2}+\tfrac{1}{2}] but d([\sqrt{2}],[0])\neq d([\sqrt{2}+\tfrac{1}{2}],[0]).
    Last edited by flyingsquirrel; September 22nd 2009 at 01:50 AM.
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  4. #4
    Super Member redsoxfan325's Avatar
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    That's the exact problem I'm running into when trying to come up with a metric for this. We need to find some invariant for all x\in[x].
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