7. a) The elements of P(A) are Ø, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}.
There are 81 ordered pairs in the relation r so they are not listed here. To give you an idea of the type of ordered pairs which occur in the relation r, the following ordered pairs are in the relation: (Ø, {1}), (Ø, {2, 3}), (Ø, {4}), ({1}, {1}), ({1}, {1, 2, 3}), ....
It was shown in question 5, that the relation r is a partial order relation, so to determine if r is a total order relation, you need only determine whether or not every pair of elements in P(A) are comparable.
b) The elements of P(B) are Ø, {1}. Hence the ordered pairs in the new relation r are (Ø, Ø), (Ø, {1}), ({1}, {1}).