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Quotient groups

  1. Aug 27, 2005 #1
    how to define R\Q?(under addition)
    R\Q={a+Q:? <a<?}
    a€R but if it is not bounded then it will repeat
    please help me
    n
     
  2. jcsd
  3. Aug 27, 2005 #2

    matt grime

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    you define [x]+[y] to be [x+y] wehre [z] means the equivalence class of y (ie the coset y+Q)
     
  4. Aug 27, 2005 #3
    quotient group is the set of cosets y+Q but if i take y€R the sets begin to repeat (it is a cyclic group, isn't it?)i
    for example R\Z ={x+Z: 0<x<1} is set of disjoint sets
    so can we also find conditions for y to be disjoint and nice?
     
  5. Aug 27, 2005 #4

    matt grime

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    of course the groups isn't cyclic, it isn't even countable os can't have a singel generator. i struggle to understand your question. what "sets" begin to repeat? plus all cosets are disjoint or equal. do you simply want a "nice" way of describing the equivalence classes? i doubt there is one.
     
  6. Aug 28, 2005 #5
    mattgrime:<do you simply want a "nice" way of describing the equivalence classes?>
    probably. sorry for misremembering the defn for cyclic groups

    what i want to know is a description for set of disjoint (but not same) cosets such that their union is real numbers.

    the R\Q question crossed my mind because of the description R\Z ={x+Z: 0<x<1} if x were bounded as 0<x<2 some of the elements of the R\Z would be same That is what i mean by "repetition".
    but i could'nt describe a set for R\Q in the same way

    All of these seemed to me related to counting. when counting real numbers(it is a bit utopian) adding a number € Z and a real number between 0 and 1 is enough. (e.g. 3.4=3+0.4 and this representation is unique using this method)
    but same method using rational numbers does not work ,
    can we say adding a number € Q and an irrational number between 0 and 1 is enough .
    (a real number is more than enough)

    thanks in advance
     
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