Question

If A is an uncountable set and B is a countable set, must A-B be uncountable?

Answer 1

Yes, $A \setminus B$ must be uncountable.

Proof. Assume $A \setminus B$ is countable. Then, we can express $A$ as $(A \setminus B) \bigcup B$. Since the union of two countable sets is countable, $A$ must be countable. This is a contradiction, and so $A \setminus B$ is uncountable.