I came across this problem which asks to show the existence of two disjoint Turing-recognizable sets $ A$ and $ B$ such that no decidable set $ C$ can separate them…

In this case, a set $ C$ is said to separate $ A$ and $ B$ if $ A \subseteq C$ and $ B \subseteq \overline{C}$ … If only $ A$ is Turing-recognizable, then we could easily set $ A$ to be $ A_{TM}$ and $ B$ as $ \overline{A_{TM}}$ . However, in this case both $ A$ and $ B$ are Turing-recognizable …. I think that $ A$ and $ B$ should be constructed using diagonalization, but could not think of a way to do it … Any help ?