Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $
- What is the cardinality of $[\pi]_S$ ?
- Prove that the quotient group $\mathbb R/S$ is uncountable.
Well I think that cardinality is zero because for all $a-b=\pi\notin\mathbb Q$ so this equivalence class is empty.
I find it strange that this quotient group is uncountable since it consists of elements only from the rational numbers and they are countable. Even with a union of all the equivalence classes we will have only $\mathbb Q$ and not $\mathbb R$.
Please share your thoughts on how to solve this.
Note: This is from set theory intro course so I probably won't understand solutions that utilize knowledge from abstract algebra, rings or group theory.