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.
Consider the elements {1, 2} and {3, 4}.
These are both elements of P(A) but under the relation r, {1,
2} is not related to {3, 4} and {3, 4} is not related to {1, 2}. Hence these two elements
are noncomparable, and so r is not a
total order relation.
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}). The relation is still reflexive, antisymmetric and transitive. Furthermore, there are only two elements in the set P(B), Ø and {1}, and these two elements are comparable. Hence the new relation r is a total order relation.