Let $$E_{TM} = \left \{ \langle M\rangle \mid L(M) = \emptyset \right\}$$
Prove that there are two languages $L_1, L_2$ such that
- $L_1, L_2 $ are infinite.
- $L_1 \cup L_2 = E_{TM}$
- $L_1 \cap L_2 = \emptyset$
- $L_1$ is decidable, $L_2$ is not recognizable.
I'm finding it really hard to find two languages that satisfy these conditions.
Especially the second condition, which two language unify to $E_{TM}$?