Practice Problems for Exam 1: Solutions 1. True or False. Justify your answer. Let A, B be two sets. 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 Collections: Mathematics