Is the above statement true? If so, how do I go about proving it?
Another thing: Given two recursively enumerable sets $Q_1$,$Q_2$, I want to prove that $Q_1\backslash Q_2$ isn't necessarily recursively enumerable. I assume that for this I need the complement of $Q_2$ to be not recursively enumerable, but could someone give a concrete example/way of of constructing a set such that I can prove my claim.
Thank you in advance.
Hint for the first question: you can assume w.l.o.g. that the recursive set is $\Bbb{N}$ (and the question needs qualifying: finite sets are recursive sets that have no non-recursive subsets, so the recursive set has to be infinite).
Hint for the second question: you can get a counter-example in which $Q_1 = \Bbb{N}$.
Both questions reduce to finding an r.e. subset of $\Bbb{N}$ that is not recursive.
If you need more details, then you need to provide some more information about what you've tried so we can know what techniques you are familiar with.