If $R,F \cap R, F \cap \overline{R}$ are regular, is $F$ regular? Let $R \in REG$. Is it true that if $F\cap R \in REG$ and
$F \cap \overline{R} \in REG$ then $F \in REG$?

I took $R = \Sigma^{\ast}$ and because $F\cap R \in REG$, $F$ must be regular. Is this the right approach?
Here's a hint. First, show that for any two sets $A, B$, that $A = (A\cap B)\cup(A\cap\overline{B})$. Then use the fact that the union of two regular languages is regular.