If $E$ and $F$ are dense uncountable sets in $\mathbb R$, then $E\cap F$ is dense in $\mathbb R$

Question

True or false:

If $E$ and $F$ are dense uncountable sets in $\mathbb R$, then $E\cap F$ is dense in $\mathbb R$.

I believe this is false. Consider two sets. The first being the non-negative rationals together with the negative irrationals. The second being the non-positive rationals together with the positive irrationals. Then each set is dense in $\mathbb{R}$ but their intersection is the singleton $0$.

Please critique my attempt to disprove this statement.

Answer 1

Your proof is absolutely correct.