 
Summary: Practice Problems for Exam 1: Solutions
1. True or False. Justify your answer. Let A, B be two sets.
(a) If A = , then B  A = B.
True. Let x B  A. By definition of set difference, we have x B and
x / A. In particular, x B. Now suppose x B. Since A = , it follows
that x / A. Thus x B  A.
(b) If B  A = B, then B = .
False. Setting B = {1} and A = provides a counterexample.
(c) If A B = A, then A B.
True. Suppose A B = A and x A. Thus, x A B also. By definition
of set intersection, x A and x B. In particular, x B.
2. Consider the three sets A = {1, 2, {4, 3}, 5}, B = {{3, 4}, 5, 1, 2}, C = {{1, 2}, {4, 3}, 5}.
(a) Which of the above sets are equal to each other?
Sets A and B are equal because they have the same elements. Set C is not
equal to A or B because 1 / C, but 1 A B.
(b) List all subsets of A.
, {1}, {2}, {{3, 4}}, {5}, {1, 2}, {1, {3, 4}}, {1, 5}, {2, {3, 4}}, {2, 5}, {{3, 4}, 5},
{1, 2, {3, 4}}, {1, 2, 5}, {1, {3, 4}, 5}, {2, {3, 4}, 5}, A.
3. Use induction to prove that for every natural number n N,
(a) 4n
