Let $X$ be a regular language, I need to prove that either $\{G \mid L(G) = X\}$ or $\{G \mid L(G) = \overline{X} \}$ is undecidable using the following hint: Use reduction to absurdity supposing that both sets are decidable and concluding that UNIVERSALITY is decidable.
UNIVERSALITY problem is $ \{ G \in CFG \mid L(G) = \Sigma^* \} $, which is known to be undecidable.
I think that the goal is to arrive to the conclusion that UNIVERSALITY is decidable, maybe with a reduction, which means that there is a contradiction in terms of decidabilty of UNIVERSALITY.