Example of recursively enumerable languages that under intersection are $\emptyset$

Question

I am trying to think about an example of a recursively enumerable languages $L_1,L_2 \in RE $ and $L_1,L_2 \notin R $ that satisfy: $L_1 \cap L_2 \in R $

I know that it will be probably something to do with $L_1 \cap L_2=\emptyset$ but I am kind of confused about the languages that can intersect in RE

Answer 1

Let $L_1$ and $L_2$ correspond to two completely different encodings of the exact same problem. Then it should be easy to make $L_1 \cap L_2 = \emptyset$.

[edit]

Let $L \in RE \setminus R$. Let $L_1 = \{0||w \mid w \in L\}$ and $L_2 = \{1||w \mid w \in L\}$ where $||$ is string concatenation. Clearly $L_1$ and $L_2$ are RE and not R, and $L_1 \cap L_2 = \emptyset$.

Answer 2

If you define, $$L_i = \{ e \ : \ \varphi_e(e)\downarrow = i \}$$ i.e. $L_i$ is the set of codes for Turing machines that halt and output $i$ when you run the machine on it's own code, then you get a family of disjoint R.E. sets, which are not recursive, indeed they are even recursively inseparable.

There are probably simpler ways, but this is the first thing that came to mind.