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.