I have two sets B which is recursively enumerable and is not recursive, and A which is recursive. Is $A-B$ recursive and / or recursively enumerable? What about $B-A$?
$B-A$ is obviously recursively enumerable (to generate its elements, I can generate B's elements and check if they are in A).
If A is the empty set or $A \cap B$ is the empty set, it's easy. Otherwise, I believe $B-A$ is not recursive (I can't tell if a number is in B, since B is not recursive) and $A-B$ is not recursively enumerable (I can generate A's elements, but I can't check if they are in B), so it's not recursive either.
Am I wrong? How can I actually (and formally) prove any of those?