So this is how i solve it but someone told me its wrong:
$B-C = B\cap \overline C $ and since $B\cap \overline C $ is r.e and B is recursive recursive sets are closed under intersection then $\overline C $ is not Recursive(because if both of them were recursive then the intersection would be recursive), also since r.e sets are closed under intersection then $\overline C $ must be r.e
and since $\overline C $ and $C$ are r.e then C is recursive
so am i wrong? is C recursive or not and why?
The argument is wrong.
Your argument is: $B\cap \bar C$ is r.e., $B$ is recursive, so $\bar C$ has to be non recursive, otherwise $B \cap \bar C$ would be recursive -- contradiction.
However, there is no contradiction: $B\cap \bar C$ can be both r.e. and recursive. This is equivalent to $B\cap \bar C$ being recursive. There is no hypothesis contradicting that.
Indeed, take $C$ to be the halting problem (r.e., and not recursive). Take $B=\emptyset$ which is recursive. Then $B\cap\bar C = \emptyset$ which is (recursive and) r.e. . So all the hypotheses are satisfied, but $C$ is not recursive.
