Given partial orders (A, R) and (B, S), a function f: A → B is called order-preserving if for all x, y ∈ A, x R y ⇒ f(x) S f(y). How many such order-preserving functions are there for each of the following, where R, S both denote ≤ (the usual "less than or equal to" relation)?
(a) A = {1, 2, 3, 4}, B = {1, 2};
(b) A = {1, ..., n}, n ≥ 1, B = {1, 2};
(c) A = {a1, a2, . . . , an} ⊂ Z+, n ≥ 1, a1 < a2 < ... < an, B = {1, 2};
(d) A = {1, 2}, B = {1, 2, 3, 4};
(e) A = {1, 2}, B = {1, . . . , n], n ≥ 1; and
(f) A = {1, 2}, B = {b1, b2,..., bn} ⊂ Z+, n ≥ 1, b1 < b2 < ... < bn.