Set Operations on Primitive Recursive Relations

Theorem
Let $$\mathcal{R}_1 \subseteq N^k$$ and $$\mathcal{R}_2 \subseteq N^k$$ be $k$-ary relations on $$N^k$$.

Let $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ be primitive recursive.

Then the following are all primitive recursive relations:
 * $$\mathcal{T} = \lnot \mathcal{R}_1$$;
 * $$\mathcal{U} = \mathcal{R}_1 \land \mathcal{R}_2$$;
 * $$\mathcal{V} = \mathcal{R}_1 \lor \mathcal{R}_2$$.

Proof
By hypothesis, the characteristic functions $$\chi_{\mathcal{R}_1}, \chi_{\mathcal{R}_2}$$ of $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ are primitive recursive.

Then we have that the characteristic functions of $$\mathcal{T}, \mathcal{U}, \mathcal{V}$$ are given by:
 * $$\chi_{\mathcal{T}} = \overline{\sgn} \left({\chi_{\mathcal{R}_1}}\right)$$;
 * $$\chi_{\mathcal{U}} = \chi_{\mathcal{R}_1} \times \chi_{\mathcal{R}_2}$$;
 * $$\chi_{\mathcal{V}} = \sgn \left({\chi_{\mathcal{R}_1} + \chi_{\mathcal{R}_2}}\right)$$.

Compare Complement of Primitive Recursive Set, Intersection of Primitive Recursive Sets and Union of Primitive Recursive Sets.

Hence the result.