such Problem 6 (20 points). Let A = (1, 2,8,90, F) be a DFA, where there exists a character a e that for all q e Q we have 8(q, a) = q, i.e. every state in A has a back-edge (loop back) for a. a. Prove by mathematical induction on n that that for all n > 0 we have 8* (q,an) = q, where an denotes a string of n characters of a. (Do not overthink! This is a simple proof that uses inductive definition of 8* provided in Slide 5 of Lecture 6) b. With the limited knowledge you have on A, Prove that either {a}* C L(A) or {a}* , L(A) = (You can make assumptions on the final/accepting states F) such Problem 6 (20 points). Let A = (1, 2,8,90, F) be a DFA, where there exists a character a e that for all q e Q we have 8(q, a) = q, i.e. every state in A has a back-edge (loop back) for a. a. Prove by mathematical induction on n that that for all n > 0 we have 8* (q,an) = q, where an denotes a string of n characters of a. (Do not overthink! This is a simple proof that uses inductive definition of 8* provided in Slide 5 of Lecture 6) b. With the limited knowledge you have on A, Prove that either {a}* C L(A) or {a}* , L(A) = (You can make assumptions on the final/accepting states F)