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?
If $R,F \cap R, F \cap \overline{R}$ are regular, is $F$ regular?
formal languages;regular languages;closure properties