# If $A ⊆ B ⊆ C$ and $A$, $C$ are decidable, then $B$ is decidable

I should prove or give a counterexample to the above statement.

In my opinion, this statement is false but I don't manage to find the right counterexample.

My idea was to define $$C = Σ^*$$ because $$Σ^*$$ is decidable and contains all the undecidable languages but I fail to find an undecidable language that contains a decidable language.

• A simple hint: have you considered finite languages? And, maybe, the smallest of them? Apr 23 at 9:49
• @Tonita for B? aren't all finite languages decidable? Apr 23 at 9:53
• So how about taking a finite language as a decidable?) Apr 23 at 9:57

2. $$A = \emptyset$$ and $$C = \Sigma^*$$ are decidable.