Question 5: Let n 2 1 be an integer and consider a set S consisting of n numbers. A function f :SS is called cool, if for all elements x of S, f(f(f(x)))=x Let An be the number of cool functions f S-S .Let f S- S be a cool function, and let x be an element of S. Prove that the set {x, f(x), f (f(x))) has size 1 or 3. . Let f S- S be a cool function, and let r and y be two distinct elements of S. Assume that f(y) = y. Prove that f(x) . . Prove that for any integer n 2 4, An-An-1+(n - 1)(n-2) An-3 Hint: Let y be the largest element in S. Some cool functions f have the property that f(y) = y, whereas some other cool functions f have the property that f(y-y