I tried to prove this using statement using the difference of sets $\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable.
In general, is it possible to say that "Let $A$, $B$ and $C$ be sets such that $B$ is not countable and $C$ is countable. If $A=B-C$, then A is countable." ?
The set of irrational numbers is no enumarable. If $\mathbb{R}$ - $\mathbb{Q}$ was enumarable, $\mathbb{R}$ would be union two countable sets which is countable.