# Decidability of a language and inclusion between two other languages

I have this assignement that asks to say if the following statement is true or false, and possibly justifying the answer:

"Let L₁, L₂ be decidable languages.
For every language L s.t. L₁ ⊆ L ⊆ L₂, L is decidable too

My first idea was to use the Halting Problem Language as L and show that is not decidable to prove the statement is false.

I tried to ask the Professor and he confirmed me that the assignment is false but he gave me a hint about explicitly describing the two languages L₁ and L₂, but I don't know which one would fit for this scope.

Thank you!

Your statement is false. Indeed, if $$L$$ is any language over $$\Sigma$$, then $$\emptyset \subseteq L \subseteq \Sigma^*.$$
• Or to generalize: when asked a question about languages in general, first ask yourself what happens for the trivial languages, the empty set and the universal set ($\Sigma^*$). – reinierpost Jun 2 '20 at 11:10