1. The empty set is a proper subset of every set. Not true because the empty set is not a proper subset of itself! is the power set of A.

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1 MAT 101 Solutions to Sample Questions for Exam 1 True or False Questions Answers: 1F, 2F, 3F, 4T, 5T, 6T, 7T 1. The empty set is a proper subset of every set. Not true because the empty set is not a proper subset of itself! 2. If 1,2,3. ( P A is the power set of A.) A, then 2 PA Not true because the elements of the power set of A are sets themselves (subsets of A) and 2 is just a number. 3. For any two finite sets A and B, if A B, then A B. Not true when A = B. 4. For any two sets A and B, the intersection of A and B is always a subset of A. True. Check this with Venn diagrams (don t forget to include the three special cases when A = B, A is a subset of B, and A and B are disjoint) 5. A set with 4 elements has exactly 10 proper subsets with cardinality greater than 1. True. Out of the 15 proper subsets of the set, 5 have to be discarded because their cardinality is 1 or less: the empty set and all 4 sets which have only one element. That leaves out exactly 10 proper subsets with cardinality greater than The set of large naturals written with 100 digits or more in decimal base (with a nonzero leading digit) is well-defined. True, it is clear which naturals belong to this set. 7. For any two finite sets A and B, A B A only if B is a subset of A. True. Again, check this with Venn diagrams considering all cases.

2 Multiple-Choice Questions Answers: 1B, 2D {1, 2, 3, 4, } is the set of naturals, or counting numbers. Consider the universal set U and the sets A and B defined below. U { x x and 2 x 18} = {2, 3, 4,, 17, 18} A = {2x x = 2, 3,, 7} = {4, 6, 8, 10, 12, 14} B = {3, 4, 5, 6, 7, 9, 12, 14, 15} 1. Which of the following is the set A B? a) {3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15} b) {4, 6, 12, 14} c) {4, 6, 8, 10, 12, 14} d) {3, 4, 5, 6, 7} 2. Which of the following is the set A B /? a) {3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15} b) {2, 11, 13} c) {2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18} d) {2, 11, 13, 16, 17, 18} Sample Problem 1 Let U a, b, c, d, e, f, g, h be the universal set and A, B, C be subsets of U defined as follows: a, b, c d, B b, e, a, d, C a, e A, Find the following:. a) A / C Answer: {a, e, f, g, h} / b) A B C Answer: {b, d} c) A C B / Answer: {c, f, g, h} / d) All proper subsets of A Answer: The 16 subsets are { }, {e}, {f}, {g}, {h}, {e, f}, {e, g}, {e, h}, {f, g}, {f, h}, {g, h}, {e, f, g}, {e, f, h}, {e, g, h}, {f, g, h}, A.

3 Sample Problem 2 Use the Venn diagram below to shade the region corresponding to the set X Y X Z Y Z. For your convenience the 8 regions in the diagram have been numbered arbitrarily. Answer Looking at the numbered regions in the Venn Diagram above, we have: Therefore, X Y = {2, 3, 4, 5} {3, 4, 7, 8} = {2, 5} X Z = {2, 3, 4, 5} {4, 5, 6, 7} = {2, 3} Y Z = {3, 4, 7, 8} {4, 5, 6, 7} = {3, 8} X Y X Z Y Z = {2, 3, 5, 8} Regions 2, 3, 5, 8 should be shaded in the Venn diagram above.

4 Sample Problem 3 a) Write a set expression for the region shaded in the Venn diagram above using any set operations. Here are two possible answers: A B B A A B A B b) Write an equivalent expression for the same region using all four set operations. [Bonus] / / Here is one possible answer: A B A B Sample Problem 4 Show the set identity / A ( A B) A B using Venn diagrams. Here you must show that the two shaded Venn diagrams associated with each side of this set expression are identical. Use a numbering of the 4 regions for the lefthand side, and then check that you have shaded the union of A and B. Sample Problem 5 Show that, in general, A ( B C) ( A B) C using Venn diagrams. Here you must show that the two shaded Venn diagrams associated with each side of this set expression are not the same. Use a numbering of the 8 regions to get these shadings.

5 Sample Problem 6 Let U, the universal set, consist of all the positive rational numbers (or fractions). p Let A be the set defined by A p, q are odd naturals and q 0 }. Find a set B that satisfied q the following 3 conditions: 1) B is a subset of the complement of A. 2) B has exactly 255 proper subsets. 3) No element of B is greater than 1. Here are two possible answers (out of infinitely many): B = {1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14, 1/16} or B = {1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9} Sample Survey Problem 1 Exercise # 10 p. 79 [Electronic Devices] DONE IN CLASS a) 496 b) 132 c) 29 d) 328 e) 470 Sample Survey Problem 2 Exercise # 12 p. 80 [Appetizers Survey] Let S be the set of surveyed customers who liked the shrimp cocktail. Then the cardinality of S is given by S 78.

6 Let M be the set of surveyed customers who liked the mozzarella sticks. Then the cardinality of M is given by M 56. Since 35 of the customers in the survey liked both types of appetizers, we have S M 35. Using the Exclusion-Inclusion Principle, we then have: S M S M S M Since 100 people were surveyed, this result implies that one customer in the survey did not like either of the two appetizers - a contradiction to the statement that everyone liked at least one of the appetizers.