Countability and uncountability of a set $A$ and the set of equivalence classes $A / R$ 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.

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.
First note that equivalence classes $R$ are subsets of $A$. Because the size of any subset of a countable set is countable, all equivalence classes are countable.

For part $3$, suppose all equivalence classes are countable. The order of a union countable sets is also countable, so the order of $A$ (which is the order of the union of all equivalence classes) is countable. This is a contradiction, so there is at least one non-countable equivalence class.
I have prove for 3. If A isn't countable and A/R is countable then there is an equivalence class that isn't countable.
We have example for it is-
If A= the set of real numbers 
Relation between a and b is either both are rational or irrational 
Then Q/R is countable having two equivalence  classes one of rationals and second is of irrationals. So here equivalence class of irrationals is uncountable.