Let $A$ be a set and $R$ an equivalence relation on $A$. Prove or disprove:
- If $A$ is countable then all the equivalence classes of $R$ are countable.
- If $A$ isn't countable then the quotient group $A/R$ isn't countable.
- If $A$ isn't countable and $A/R$ is countable, then there is an equivalence class in $R$ that isn't countable.
Well, I thought of going from the definitions and say that two equivalence class is either identical or foreign, also, the equivalence relation (that is transitivity, reflexivity and symmetry) can add only a finite number of elements so at most it will be countable. So it's true.
False: $A=\Bbb R , \ R=mod (2) $ so there are only two equivalence class and the quotient group consists of only $1$ and $0$.
I'm pretty sure it's true, $A/R$ at most has the cardinality $\aleph_0$ and it's possible to have an equivalence class that is in one to one correspondence with $A$. But I don't really have an idea on how to prove it.
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.