TITLE: Uncountable minus countable set is uncountable
QUESTION [9 upvotes]: Problem statement
Prove that if $A$ is an uncountable set and $B$ is a countable set, then $A\setminus B$ must be uncountable.
What I think
The statement does not mention $A$ and $B$ relationship. I think there are two possibilities:

If $A \cap B = \emptyset $, then $A\setminus B$ is trivially uncountable
If $A \cap B = B$, then $B \subset A$ and as a bijection can not be made between $A\setminus B$ and $\mathbb{N}$, $A\setminus B$ is uncountable.

And there is where I'm stuck. How can I prove that a bijecton can't be made?
TIA.

REPLY [3 votes]: Suppose that $A-B=A-A\cap B$ is countable.  Then you have a bijection between $A-B$ and the set of even natural numbers. Since $B$ is countable then $A\cap B$ is also countable and you have a bijection between $A\cap B$ and the set of odd natural numbers.
Now, taking the union of those bijections, you get the bijection between the set
$A=(A-B)\cup (A\cap B)$ and the set of natural numbers, which is an absurd because $A$ is uncountable. Thus $A-B$ is uncountable. Q.E.D.
If $A\cap B$ is just finite countable then the proof goes similarly with obvious changes, see comments.