2. (25%) Let S = {a,b}, and let f:9* ** be the function that replaces each a in the input by ab, and each b by a. For example, f(abb) = abaa, and f(e) = E. (a) Define f by recurrence on *. (b) Prove by induction on I* that #a(f(x)) = || for all we *. 2. (25%) Let S = {a,b}, and let f:9* ** be the function that replaces each a in the input by ab, and each b by a. For example, f(abb) = abaa, and f(e) = E. (a) Define f by recurrence on *. (b) Prove by induction on I* that #a(f(x)) = || for all we *