this is a computer science question

Problem 6 (20 points). Let A = (Q, 9, 8, 90, F) be a DFA, where there exists a character ae that for all q e Q we have (q, a) = q, i.e. every state in A has a back-edge (loop back) for a. such a. Prove by mathematical induction on n that that for all n > 0 we have 8* (q,a") = 4, where a" 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) (A) = 0 b. With the limited knowledge you have on A, Prove that either {a}* = C(A) or {a}* n (You can make assumptions on the final/accepting states F) Problem 6 (20 points). Let A = (Q, 9, 8, 90, F) be a DFA, where there exists a character ae that for all q e Q we have (q, a) = q, i.e. every state in A has a back-edge (loop back) for a. such a. Prove by mathematical induction on n that that for all n > 0 we have 8* (q,a") = 4, where a" 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) (A) = 0 b. With the limited knowledge you have on A, Prove that either {a}* = C(A) or {a}* n (You can make assumptions on the final/accepting states F)