Please do any three parts. It is related to recursion and induction. Thank you so much!

EExercise 117. Carry out the following for the rev operator. (a) Calculate rev(1, 0) step by step. (b) Show that for every list l over an arbitrary set len revl lenl. Hint: You may want to use a statement from a previous assessed exercise to help with this. (c) Show that rev(l #1') =revl' # revl. (d) Show that rev revl=l for all lists l over S. Hint: You may want to show rev[s] = [s] separately. (e) Use the rev operator to give an alternative definition of the concatenation operator which uses recursion over the second argument. Give an argument that your definition agrees with the original. Justify each step in your proofs. EExercise 117. Carry out the following for the rev operator. (a) Calculate rev(1, 0) step by step. (b) Show that for every list l over an arbitrary set len revl lenl. Hint: You may want to use a statement from a previous assessed exercise to help with this. (c) Show that rev(l #1') =revl' # revl. (d) Show that rev revl=l for all lists l over S. Hint: You may want to show rev[s] = [s] separately. (e) Use the rev operator to give an alternative definition of the concatenation operator which uses recursion over the second argument. Give an argument that your definition agrees with the original. Justify each step in your proofs