Countability of the difference of two uncountable sets

Question

If $A$ and $B$ are two uncountable subsets of $\mathbb R$ then the set $A\setminus B$

• (1) has to be uncountable
• (2) has to be countable
• (3) has to be infinite
• (4) None of the above.

Answer 1

It can be any. If $A=(0,1)$ and $B=(0,\frac{1}{2})$, then $A\setminus B=[\frac{1}{2},1)$ which is uncountable and hence infinite.

If $A=[0,1]$ and $B=(0,1)$, then $A\setminus B$ is finite, and hence countable.

So, if $A$ and $B$ are uncountable subsets of $\mathbb{R}$, then their difference $A\setminus B$ may be empty, finite, countable or uncountable. So the correct answer is (d), none of the above as it doesn't HAVE to be any of them.