Let S = {c, o, u, n, t, m, e, 2}. Observe |S| = 8. (a) How many total functions are there f:S → S if there are no restrictions? (b) How many partial functions are there f:S → S if there are no restrictions? (c) How many total functions are there f: S→ S with restrictions that f(m) is a vowel and f(u) is not a vowel? Recall the vowels are a, e, i, o and u.

Let f be a total or partial function f:S -> T where S = {d,i,s,c,r,e,t} and T = {1,2,3,4,5,6,7} How many partial functions f:S -> T are there if f(d) and f(I) are not even? How many partial functions are if f(d) exists in {2,3,4,5} and f(I) is even? How many injective functions f: S -> T are there if f(d), f(I), and f(s) are all elements of {1,3,4}?

For a derived varaible f : states → R is strictly decreasing if and only if for states q, q' : q → ď = f(d) > f(q) O q + d = f(d) > f(q) O q → ɖ = f(d) f(q)

Let S={1,2,3,4,5}S={1,2,3,4,5} and T={1,2,3,4,5,6,7,8}T={1,2,3,4,5,6,7,8}, so |S|=5|S|=5 and |T|=8|T|=8. Consider total and partial functions, f:S→Tf:S→T.