3. [Hashing, 10+10 points] Let U denote the universe [0,,N1]. A function h:U[0,,m1] is a perfect hash function for SU if h(x)=h(y) for any x,yS,x=y. A set H of such functions is said to be (N,m,n)-perfect if for every SU with S=n, there exists hH such that h is perfect for S. Let H={h1,,ht} be a (N,m,n)-perfect set. Starting with U0=U, construct Ui+1= Uihi+11(j) for i