(a) Reversing a string Let & be any alphabet, and let w be a string over 2. The reverse of w, denoted by wR, is defined recursively as: R = e, (wa) = a(w), where a 2 is a character. In English, (wa)R = a(wk) means that reversing the string wa is the same as the concatenation of the symbol a and the string wR. Prove by induction that (xy)R = y xR for all strings x, y over the alphabet E. (b) Some properties of palindromes A palindrome is a string w (over an alphabeta E) that is the same as its reverse: w = wR iff w is a palindrome. Let L = {u(vR)|u, v are strings over 2 and v is a nonempty prefix of u} U EU {f}. Show that: The set of all palindromes is a subset of L. If |S| > 1, then there exists an object that belongs to L but does not belong to the set of palindromes. In other words, the set of all palindromes is a proper subset of L. You can use the results from Part (a) for this question. We say that A is a proper subset of B if A S B and there exists x B such that x A