# Prove that $A \oplus B$ is recursive if $A$ and $B$ are recursive

Let define $$A \oplus B = \{2x \mid x \in A\} \cup \{2x + 1 \mid x \in B\}$$. Prove that $$A \oplus B$$ is recursive if $$A$$ and $$B$$ are recursive.

I am currently having the following idea.

Since $$A$$ is recursive, $$P_A$$ the characteristic of $$A$$ is recursive. Similarly, $$P_B$$ is recursive.

So, we need to show that $$P_{A \oplus B}$$ is recursive.

How can I define the characteristic predicate of $$A \oplus B$$ in term of $$P_A$$ and $$P_B$$?