I am trying to find $A/\sim$, a set $A$ with an equivalence relation $\sim$ such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.
I already tried with $A = \Bbb{R}$ and $x \sim y := x – y \in \Bbb{Z}$. The equivalence classes are the sets of reals with the same fractional part. You can show that $A/ \sim = [0,1)$. Which is easy to show is uncountable with an injection $f: \{0,1\}^{\infty} \to [0,1)$ (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?
Best Answer
You are right that you have uncountably many equivalence classes, but each equivalence class is only countably infinite.
It is easier to find relations with uncountably many uncountably large classes on the plane $\Bbb R^2$.
It might also be easier to go the "other way". Build the equivalence classes first as an uncountable collection of uncountable, disjoint sets, and define the relation as "Two elements are related iff they are in the same set."