Discrete Structures

– You must justify your answers.
– The answers should be concise, clear and neat.
– When presenting proofs, every step should be justified.

Question 5: Let n2 1 be an integer and consider a set S consisting of n numbers. A function f:SSis called cool, if for all el

Question 5: Let n2 1 be an integer and consider a set S consisting of n numbers. A function f:SSis called cool, if for all elements r of S Let An be the number of cool functions f:S-S Let f : S → S be a cool function, and let r be an element of S. Prove that the set has size 1 or 3. Let f : S → S be a cool function, and let x and y be two distinct elements of S. Assume that f()-y. Prove that f(x) y. . Prove that for any integer n 2 4, An = An-l + (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. Show transcribed image text