Practice Exam #1 September 14, 2012

Transcription

1 Lenarz Math 102 Practice Exam #1 September 14, 2012 Name: Directions: This practice exam is longer than the in-class exam. It is meant to give you an idea of the type and difficulty level of questions you might see on the exam. 1. Explain the difference between (a) and { }. is the symbol for the empty set. It is a set with no elements. But { } is the set that contains the empty set. It is not empty as it has one element. (b) the symbols and. The symbol is used for denoting a subset and the symbol is used for denoting a proper subset. For example, if we write A B, this means that A is a subset of B (and it is possible that A = B). If we write A B, this means that A is a proper subset of B (i.e. A is a subset of B and A B). 2. Find n(a) for the following sets: (a) A = {1, 3, 46, 100, 3000} n(a) = 5 (b) A = {{0}, {1, 2}, {3, 4, 5}} n(a) = 3 (c) A = {1, {1}, {1, 2}} n(a) = 3 3. Determine if the sets are equal, equivalent, both or neither. (a) A = {1, 2}, B = {1, 2, 3} The sets are not equal because 3 B and 3 / A. The sets are not equivalent because n(a) = 2 but n(b) = 3.

2 Math 102 September 14, 2012 Page 2 (b) A = {a, b, c}, B = {c, a, b} The sets are equal, because every element of A is in B and vice versa. They are equivalent because n(a) = 3 = n(b). (c) A = {1, 3, 5}, B = {3, 5, 7} The sets are not equal because 1 A but 1 / B (could also refer to 7 B but 7 / A). The sets are equivalent because n(a) = 3 = n(b). 4. List all of the two-element subsets of the set {1, 4, 8, 10}. {1, 4}, {1, 8}, {1, 10}, {4, 8}, {4, 10}, {8, 10} 5. Let = {q, r, s, t, u, v, w, x, y, z}, A = {q, s, u, w, y}, B = {q, s, y, z}, = {v, w, x, y, z}. Express the following sets using the listing/roster method (a) A (B ) Note that B = {y, z}, so we have A (B ) = {q, s, u, w, y, z} (b) A Note that A = {r, t, v, x, z}, so we have A = {w, x, y} 6. Let = {all sports}, A = {all outdoor sports}, B = {all sports using a ball}, = {all water sports}, D = {all sports requiring a helmet}. Describe the following sets in words (a) A B D A B D is the set of all outdoor sports that use a ball and require a helmet.

3 Math 102 September 14, 2012 Page 3 (b) A A is the set of all water sports that are not played outdoors (that is, all indoor water sports). (c) A (B ) A (B ) is the set of all outdoor sports that are either water sports or use a ball. 7. Name the shaded region using correct set theory notation. (a). A B (A B) or (A B ) would be possible answers. (b). A B B (A ) or A B would be possible answers.

4 Math 102 September 14, 2012 Page 4 8. se the Venn diagram below to find A 10 B (a) n(a) n(a) = = 25 (b) n(a B) n(a B) = = 14 (c) n(b ) n(b ) = = 24 (d) n( A) n( A) = = 1 (e) n( ) n( ) = = 26

5 Math 102 September 14, 2012 Page 5 (f) n(a B ) n(a B ) = 1 9. Given the following n(a B ) = 28 n(a B) = 8 n(a ) = 9 n(a B) = 8 n( B) = 11 n(a B ) = 6 n( (A B)) = 1 find (a) n(a) (b) n(b) (c) n() The following Venn diagram represents the situation A 2 B ? (a) (b) (c) n(a) = = 16 n(b) = = 19 n() = = 15

6 Math 102 September 14, 2012 Page On a recent Valentine s Day, 86 husbands were interviewed, and 15 said they were going to buy roses for their wives, 25 said they were going to take their wife out for dinner, and 18 said they were going to buy their wives candy. Of these, 8 said they were going to buy roses and take their wives out to dinner. Five said they were going to buy both roses and candy, and 11 said they were going to buy their wives candy and take them out to dinner. Only 2 men said they were going to treat their wives to all three roses, candy, and dinner. How many men are going to buy roses and take their wives out for dinner, but not buy them candy? Let D be the set of men who will take their wives to dinner, R the set of men who will buy their wives roses, and the set of men who will buy their wives candy. The following Venn diagram represents the situation D 6 R So the number of men who will buy roses and take their wives to dinner but not buy them candy is n(d R ) = 6

7 Math 102 September 14, 2012 Page A survey of 100 people showed the following 2 own stocks, bonds, and Ds 6 own stocks and bonds 5 own stocks and Ds 3 own bonds and Ds 15 own stocks 12 own bonds 15 own Ds How many investors own bonds or Ds (or both) but no stocks? Let S be the set of people surveyed who own stocks, B the set of people surveyed who own bonds, and the set of people surveyed who own Ds. The following Venn diagram represents the situation S 4 B So the number of investors who own bonds or Ds (or both) but no stocks is n((b ) S ) = = 15