Finding languages such that $L_{1} \subseteq L_{2} \subseteq L_{3}$ where $L_{1}, L_{3} \notin \mathbb{R}$, $L_{2} \in \mathbb{R}$

Question

I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin \mathbb{R}$ and $L_{2} \in \mathbb{R}$.

I know they exist, I need help finding them.

Answer 1

Any language $L$ over a finite alphabet $\Sigma$ is both contained in some regular language and in some non-regular language.

Firstly, $\Sigma^*$ is regular and contains the language.

Secondly, if $x,y\not\in\Sigma$, then $\{sx^ny^n:s\in L,n\in\mathbb{N}\}$ is non-regular and contains $L$. The standard proof that $\{x^ny^n:n\in\mathbb{N}\}$ is non-regular using the pumping lemma can be copied completely to prove this.