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 $P_B$?