This exercise was taken from the book “Languages and Machines: An Introduction to the Theory of Computation” by Thomas Sudkamp. It refers to exercise 12 (b) chapter 12. Given a language L which is recursive enumerable, I have to prove that the following property is undecidable:

The text says that it is sufficient to prove that it is a non trivial property.

I tried to solve the exercise as follow:

Consider the empty language $ \emptyset$ , which contains only the empty string λ, in other words $ \emptyset$ = {$ \lambda$ }. Then $ \emptyset^-$ which is the negation of the empty set, will contains some string which is not $ \lambda$ . By doing this, I’ve found a language which is finite and does satisfy the property, but I’ve also found another language which doesn’t satisfy the property because $ \emptyset^-$ it is not finite. In conclusion the property it is not trivial, and by the Rice’s theorem it is impossible to decide that property.

I’m not sure if I’m doing the right thing here and I haven’t found any solution to this exercise… Can anyone help or at least tell me if I’m doing it right?

Thank you very much.