If $A$ is dense in $\Bbb R$, then $A∩Q$ is dense in $\Bbb R$ or $A∩(\Bbb R\setminus Q)$ is dense in $\Bbb R$

Question

So I got this on an assignment and I can't find a way to prove or disprove the statement:

If $A$ is dense in $\Bbb R$, then $A∩Q$ is dense in $\Bbb R$ or $A∩(\Bbb R\setminus Q)$ is dense in $\Bbb R$.

I'm cetain it has something to do with the density of Rational/Irrational numbers, but I can't figure it out.

Answer 1

The statement is wrong.

$A= \left((-\infty , 0) \cap \mathbb Q)\right) \cup (0, \infty) \cap \left( \mathbb R \setminus \mathbb Q)\right)$ is dense in $\mathbb R$. However none of $A \cap \mathbb Q$ nor $A \cap(\mathbb R \setminus \mathbb Q)$ is.