Union of Primitive Recursive Sets

Theorem
Let $$A, B \subseteq \N$$ be subsets of the set of natural numbers $$\N$$.

Let $$A$$ and $$B$$ both be primitive recursive.

Then $$A \cup B$$, the union of $$A$$ and $$B$$, is primitive recursive.

Proof
$$A$$ and $$B$$ are primitive recursive, therefore so are their characteristic functions $$\chi_A$$ and $$\chi_B$$.

Let $$n \in \N$$ be a natural number.

Then $$n \in A \cup B \iff \chi_A \left({n}\right) + \chi_B \left({n}\right) > 0$$.

So:

$$ $$

Thus $$A \cup B$$ is defined by substitution from the primitive recursive functions $$\sgn$$, $$\operatorname{add}$$, $$\chi_A$$ and $$\chi_B$$.

Hence the result.