Show that the following problem is decidable I must show that the following problem is decidable:  Given $ \Sigma = \{a,b\}$ and $\alpha$ a regular expression, is it true that the language defined by $\alpha$ contains all the odd-length strings in $ \Sigma^*$ but no string consisting only of a's?  ($\varepsilon$ is assumed to consist only of a's.)
I would say the answer to the question is false, however I don't know how to show that it is decidable.
From what I understand, I can determine whether a problem is decidable based on whether it finishes execution or runs in an infinite loop forever. What I don't understand however is the context of this problem, since it is not an actual program. I feel like there is not enough information here to draw a conclusion. There is something related to decision procedures required to solve this (Emptiness, Totality, etc not sure which one however). How can I determine whether this problem is decidable?

Yes, it is decidable, because no language can contain all the odd-length strings in $ \Sigma^*$ but no string consisting only of a's.
