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?
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.
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)