# Show that a set is turing reducible to another set

Consider an operator $$+$$ defined on $$P(N)$$ as follows: $$A + B = \{2x\mid x \in A\}\cup \{2x + 1\mid x \in B\}$$

Show that both $$A$$ and $$B$$ are Turing-reducible to $$A+B$$

Does it mean that There is a turing machine which converts an input from $$A$$ to an element of $$B$$
on the tape. There is a term called oracle that also pops up. So if we have an oracle for $$B$$
does it mean that there is a decider for $$B$$? What's an intuitive workable definition of oracle?

The definition of Turing reducible, is that there exists a Turing reduction - an algorithm for $$A+B$$ that uses an oracle to $$A$$.
To understand this, you must first understand oracle machines. Those machines, are like turing machines, but can ask an "oracle" whether any $$x$$ is in $$A$$ or not, and would get an answer in $$O(1)$$, regardless of what $$A$$ is (doesn't even matter if it is not decidable!)
So, a turing reduction basically means that if you know for any $$x$$ whether its in $$A$$ or not in $$A$$, then you can decide whether a given $$x$$ is in $$A+B$$. Formally, this statement can be written as $${(A+B)}^A\in R$$.
So basically the question asks you to show that $${(A+B)}^A,{(A+B)}^B\in R$$