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 $$ \begin{align*} A^c &= (A^c \cap B^c) \cup (A^c \cap B) \\ &= (A \cup B)^c \cup (B \cap (A^c \cup B^c)) \\ &= (A \cup B)^c \cup (B \cap (A \cap B)^c). \end{align*} $$ Using this you can recursively enumerate $A^c$. I'll let you figure out how.

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  • $\begingroup$ Of course! By negation theorem $(A \cup B)^c, B$ and $(A \cap B)^c$ are r.e., thus $A^c$ is too because is union and intersecction of r.e. Thank you! $\endgroup$ – Tom Ryddle Oct 4 '18 at 3:56

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