(15 pt.) Given a DFA M=(Q,,,q0,F) that recognizes a regular language A, construct an NFA N=(Q,,,q0,F) such that: - The states of N are the same as the states of M. - The start state of N is the same as the start state of M. - F=F{q0}. That is, the accepting states of N are the accepting states of M and the start state. - is defined as follows for any qQ and any c : (q,c)={(q,c)(q,c){q0}ifq/Forc=ifqFandc= That is, the transitions of N are the transitions of M and -transitions from the accepting states of M to the start state. a. (10 pt.) Show, by giving a counterexample, that this construction fails to prove that the class of regular languages is closed under Kleene star. That is, show that if M is a DFA that recognizes a regular language A, then this construction may not result in an NFA N that recognizes the Kleene star of A. Justify your answer. Hint: give a specific DFA M=(Q,,,q0,F) for which the constructed NFA N=(Q,,,q0,F) does not recognize the Kleene star of the language of M. b. (5 pt.) Explain how to fix this construction to prove that the class of regular languages is closed under Kleene star