Which of the following statements are true for all strings x, y, and z over ? For those that are true, give direct proofs. For those that are false, give counterexamples. (Assume that has at least two symbols, say, 0 and 1, and possibly others. Recall that denotes the empty string over any alphabet.) (a) is a prefix of x (b) z is a prefix of x. (c) If x is a prefix of y, then y is a prefix of x. f x is a prefix of y, then y is a suffix of x. (e) If x is a prefix of y and y is a prefix of x, then y (f) If x is a prefix of y and y is a prefix of z, then x is a prefix of z. (g) x has the same number of prefixes as suffixes. (h) If x and y are both prefixes of z, then either x is a prefix of y or y is a prefix of x. (i) If x and y are both substrings of z, then either x is a substring of y or y is a substring of x