If $A,B$ are r.e. and $A\cup B,A \cap B$ are recursive, then so are $A,B$ Let be $A, B \subset \mathbb{N}$ are recursively enumerable, $A\cup B$ and $A \cap B$ recursive. I want to show that $A$ and $B$ are recursive. 

By negation theorem $X \subset \mathbb{N}$ is recursive iff $X$ and $X^c$ are recursively enumerable. 

How do I complete the proof using that?
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.