Is $\mathbb{R}\setminus\mathbb{Q}$ countable?

Question

I need to prove that $\mathbb{R}\setminus\mathbb{Q}$ is countable or uncountable. I believe it is uncountable. I am not sure how to prove it. $\mathbb{R}$ is known to be uncountable and $\mathbb{Q}$ is countable. By reason when I take the difference of the two it would be uncountable. How do I prove this?

Answer 1

If $\mathbb{R} \setminus \mathbb{Q}$ is countalbe, then $\mathbb{R}$ is countable. A contradiction.