Suppose f is a computable function from a recursively enumerable set U to the natural numbers and that L,K are r.e. subsets of U. Is f(L-K) a difference of r.e. subsets? The motivation comes from


[Primes occurring as orders of elements of a finitely presented group](https://mathoverflow.net/questions/121178/primes-occurring-as-orders-of-elements-of-a-finitely-presented-group)


A positive answer would mean that the theorem proposed in HW's nice answer is 100% correct. Otherwise the $\epsilon$-clarification in my answer is actually needed. 



Computable images of differences of r.e. sets
Is f injective? If so, yes. If not, no. In the second case, you could achieve any $\Sigma^0\_2$ set.


