I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, I need to prove that the group of all equivalence classes is uncountable. I have an idea how to move from step one (proving it's $\aleph_0$) to step two (proving that the group of all equivalence classes is uncountable), by using the laws of addition in cardinality. However, I have no idea how to even approach the first step. Any help or idea will be appreciated!
Thanks a bunch!