Reversed Language of a Non Regular Language

Question

Is the following saying true or false? In any case why?

Thanks!

Answer 1

Hint:

• Consider the following language: $$ X = \{ a^pb\Sigma^* \mid p \text{ is a prime number} \} \cup \{\Sigma^*ba^q \mid q \text{ is not a prime number}\}. $$
• Is it regular or non-regular?
• Can you find an element $x \notin X$? How $x^R$ does look like?
• Can you modify it so it would have the desired property?
• Intuitively harder, but technically simpler version can be made of the complement of language $Y = \{a^nb^n \mid n \in \mathbb{N}\}$.

I hope this helps $\ddot\smile$