Is the set $A = [0,1]\setminus\mathbb{Q}$ countable or not?
What I am thinking is $A$ consist of irrational numbers in the interval $[0,1]$ hence it is subset of irrational numbers. As set of irrational numbers is uncountable so I think set $A$ is also uncountable.
Best Answer
Yes, it is uncountable, but not for that reason. For instance, $\left\{\sqrt2+n\,\middle|\, n\in\Bbb N\right\}$ is also a set of irrational numbers, but it is countable.
However, if $[0,1]\setminus\Bbb Q$ was countable, then, since $\Bbb Q\cap[0,1]$ is countable, $[0,1]$ would be countable too, since it's the union of them.