I am trying to find , a set A with an equivalence relation such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.
I already tried with and . The equivalence classes are the sets of reals with the same fractional part. You can show that . Which is easy to show is uncountable with an injection (this is the standard method I use to show a set is uncountable).
However, I think all elements in my equivalence classes are countable, as they are all natural numbers plus a specific fractional part. Am I right to think like this, does it make my example false ? In case it is false, is there a way to "fix" it?