Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Question

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.

Thanks.

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.

Answer 1

Note that equivalence classes ARE NEVER EMPTY. If anything $\pi\in[\pi]_S$. You can show that $[x]_S$ is always countable by construction a bijection between $[x]_S$ and $\Bbb Q$.

The quotient set (also group, but you mentioned that you don't want algebraic arguments) is not made of rational numbers, but rather of an element from each equivalence class. By knowing that the equivalence classes are countable, you can easily prove that there must be uncountably of them, because their union is $\Bbb R$.