# How to show that these two disjoint sets $A$ and $B$ exist

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 ?

• You can find some example on Wikipedia. Jul 1 '20 at 9:57