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Math Help - Equivalence classes

  1. #1
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    Equivalence classes

    This is a question from my professor that I couldnt understand.
    Let X=R, x~y if x-y is a subset of Z(integers).
    Describe the set of equivalence classes.
    He said it's just a circle.
    I have no idea what he meant.
    Anyone can understand what he meant?
    THanks.
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  2. #2
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    Quote Originally Posted by ninano1205 View Post
    This is a question from my professor that I couldnt understand.
    Let X=R, x~y if x-y is a subset of Z(integers).
    Describe the set of equivalence classes.
    Just what are you saying, "x-y is a subset of Z".
    Are you saying that the difference is a integer?
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  3. #3
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    subset of integers

    is integers?? right?
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  4. #4
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    Quote Originally Posted by ninano1205 View Post
    is integers?? right?
    I thought as much. In that case, the mention of circle has no relevance.
    As an aside, this fits into a different problem that I have worked recently.
    Now one equivalence class is simply the set of integers: \left[ 0 \right] = \mathbb{Z}.

    Do you know about the floor function?
    \left\lfloor z \right\rfloor is the greatest integer not exceeding z.
    We can show that this function is well defined on \Re having the property that:
    \left\lfloor x \right\rfloor  \leqslant x < \left\lfloor x \right\rfloor  + 1\, \Rightarrow \,0 \leqslant x - \left\lfloor x \right\rfloor  < 1.

    Now back to your question. Define a function on \Re as \mathbb{D}(x) = x - \left\lfloor x \right\rfloor (decimal part) .
    Take note that \left( {\forall x \in \Re } \right)\left[ {0 \leqslant \mathbb{D}(x) < 1} \right].
    For the general equivalence classes: \left( {\forall x \in \Re } \right)\left[ {\left[ x \right] = \bigcup\limits_{k \in \mathbb{Z}} {\left\{ {\mathbb{D}(x) + k} \right\}} } \right].
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