Complement of Primitive Recursive Set

Theorem
Let $$S \subseteq \N$$ be primitive recursive.

Then its relative complement $$\N - S$$ of $$S$$ in $$\N$$ is primitive recursive.

Proof
By definition, we have that the characteristic function $$\chi_{\N - S} \left({n}\right) = 1$$ iff $$\chi_{S} \left({n}\right) = 0$$.

So $$\chi_{\N - S} \left({n}\right) = \chi_{\left\{{0}\right\}} \left({\chi_{S} \left({n}\right)}\right)$$.

Thus $$\chi_{\N - S}$$ is obtained by substitution from $$\chi_{\left\{{0}\right\}}$$ and $$\chi_{S}$$.

The result follows from Set Containing Only Zero is Primitive Recursive.