Cardinality of rational number and irrational number

Question

Since there is a rational number between any two irrational numbers, so can we say that rational number and irrational number have the same cardinality?

Answer 1

Of course NOT.

The argument is not correct. For example, I can say there always exists a rational number between any two distinct real numbers, but apparently $\mathbb{Q}$ and $\mathbb{R}$ don't have the same cardinality.

Answer 2

No. $\mathbb Q$ is countable. Suppose that the irrational numbers have the same cardinality, then $ \mathbb R \setminus \mathbb Q$ is also countable. This gives

$ \mathbb R = \mathbb Q \cup (\mathbb R \setminus \mathbb Q)$ is countable, a contradiction.