Problem 14.9. Let X and Y be finite sets. (a) How many binary relations from X to Y are there? (b) Define a bijection between the set [XY] of all total functions from X to Y and the set YX. (Recall Yn is the Cartesian product of Y with itself n times.) Based on that, what is [XY] ? (c) Using the previous part, how many functions, not necessarily total, are there from X to Y ? How does the fraction of functions vs. total functions grow as the size of X grows? Is it O(1),O(X),O(2X), ? (d) Show a bijection between the powerset, pow(X), and the set [X{0,1}] of 0-1-valued total functions on X