18 Let 2 be a nonempty set. Define on P(2) the following relation 3: for any A, B E P(), A 3 B if and only if cardA card B. Show that (P, 3) is a direct order. Hint. For [R], show that there is relevant function from set A to itself. (T) of (P(2), 3). Suppose that A B and BC. Then, there are injective (one-to-one) functions [A, B, f) and (B,C,g). Show that (A,C, g f] is injective. (SL) Show that given any two sets A and B, there is a set of larger or equal cardinality that either A and B. 19 Under the conditions of Problem 18, is (P(2), 3) a direct order? 18 Let 2 be a nonempty set. Define on P(2) the following relation 3: for any A, B E P(), A 3 B if and only if cardA card B. Show that (P, 3) is a direct order. Hint. For [R], show that there is relevant function from set A to itself. (T) of (P(2), 3). Suppose that A B and BC. Then, there are injective (one-to-one) functions [A, B, f) and (B,C,g). Show that (A,C, g f] is injective. (SL) Show that given any two sets A and B, there is a set of larger or equal cardinality that either A and B. 19 Under the conditions of Problem 18, is (P(2), 3) a direct order