If $A,B$ are r.e. and $A\cup B,A \cap B$ are recursive, then so are $A,B$
The basic observation is
$$
\begin{align*}
A^c &amp;= (A^c \cap B^c) \cup (A^c \cap B) \\ &amp;=
(A \cup B)^c \cup (B \cap (A^c \cup B^c)) \\ &amp;=
(A \cup B)^c \cup (B \cap (A \cap B)^c).
\end{align*}
$$
Using this you can recursively enumerate $A^c$. I'll let you figure out how.