If $x \in [0, 1]$, what is $\text{P}(x\in \mathbb Q)$? In other words, what is the probability that $x$ is rational?

This is what I tried:

$$\begin{array}{rcl}\text{P}(x \in \mathbb Q) &=& \displaystyle \int^1_0 f(x)\,dx \end{array}$$

where

$$f(x) = \begin{cases}1, & x \in \mathbb Q\\0, & x\notin\mathbb Q\end{cases}$$

However, the function is not Riemann-integrable.

I want to try comparing the cardinalities of the rational set and the irrational set by using a one-to-one mapping between the two sets. But I don't have idea how I can do it. Can anyone give a hint?