Subset of unary recursive functions is not recursive if it and its complement are non empty.

Question

Suppose $S$ is a subset of all unary recursive functions such that $S$ and its complement are non-empty. Letting $f_{n,1}$ denote the unary function computed by program $P_n$ (if it exists). How do I show that $$ \{n\in\mathbb{N}:f_{n,1}\in S\} $$ is not recursive?

Answer 1

Apply Rice's Theorem. This is a direct application.