Show that the class of regular languages over an alphabet sigma is closed under the operation Even Only (L), defined as EvenOnly(L) = {w belongsto L such that |w| is even} Your proof should have three stages: Setup Consider an arbitrary DFA M = (Q, sigma, delta, q_0, F), and call the language of this DFA L. Construction Build a new DFA whose language is EvenOnly(L). To do so, fill in the blanks M' = (Q', sigma, delta', q', F') where Q' = _________ This will be the set of states for your new machine. Delta'(______, x) = For each possible input to the transition function, specify the output. The blank input is a state in Q' and x belongs sigma. q' = _________ What is the initial state of M'? Make sure you choose an element of Q'. F' = _________ What is the set of accepting states of M'? Choose a subset of Q'. Justification To prove that the construction of correct, you will need to prove that L(M') = EvenOnly(L) for any L. Fix an arbitrary but unknown language L. Prove two things: (1) Assume that some string, call it w, is accepted by M'. Prove that w is in EvenOnly(L). (2) Assume that some string, call it y, is in EvenOnly(L). Prove that y is accepted by M