1) The set of the days of the week A) {Saturday, Sunday} B) {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Sunday}

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1 Review for Exam 1 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. List the elements in the set. 1) The set of the days of the week 1) A) {Saturday, Sunday} B) {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Sunday} C) {Friday, Monday, Saturday, Sunday, Thursday, Tuesday, Wednesday} D) {Tuesday, Thursday} Write a word description of the set. 2) {January, February, March, April, May, June, July, August, September, October, November, 2) December} A) months of the year B) seasons of the year C) days of the year D) days of the week Determine if the set is the empty set. 3) {0, } 3) A) Yes, it is the empty set. B) No, it is not the empty set. 4) 4) {x x < 6 and x > 10} A) Yes, it is the empty set. B) No, it is not the empty set. Determine whether the statement is true or false. 5) 7 {1, 2, 3,..., 40} 5) A) True B) False Fill in the blank with either or to make the statement true. 6) Manitoba the set of states in the United States 6) A) B) Express the set using the roster method. 7) the set of odd natural numbers less than 21 7) A) {0, 1, 3, 5,..., 19} B) {1, 3, 5,..., 19} C) {1, 3, 5,..., 21} D) {2, 4, 6,..., 20} 8) 8) {x x N and x is greater than 13} A) {14,16,18,...} B) {13,14,15,...} C) {14,15,16} D) {14,15,16,...} Make sure you know what symbolic forms are.

2 Find the cardinal number for the set. 9) {x x is a day of the week that begins with the letter N} 9) A) 2 B) 3 C) 1 D) 0 Are the sets equivalent? 10) A is the set of residents age 68 or older living in the United States 10) B is the set of residents age 68 or older registered to vote in the United States A) Yes B) No 11) 11) A = {7, 8, 9, 10, 11} B = {6, 7, 8, 9, 10} A) Yes B) No Determine whether the set is finite or infinite. 12) {x x N and x 1000} 12) A) Finite B) Infinite 13) 13) {x x N and x 100} A) Finite B) Infinite 14) 14) The set of natural numbers less than 1 A) Finite B) Infinite Are the sets equal? 15) {28, 30, 32, 34, 36} = {30, 32, 34, 36} 15) A) Yes B) No Determine whether the statement is true or false. 16) Ted {Bob, Carol, Ted, Alice} 16) A) True B) False 17) 17) {Carol} {Bob, Carol, Ted, Alice} A) True B) False Use,,, or both and to make a true statement. 18) {a, b} {z, a, y, b, x, c} 18) A) B) C) D) and Determine whether the statement is true or false. 19) {France, Germany, Switzerland} 19) A) True B) False 2

3 List all the subsets of the given set. 20) {Siamese, domestic shorthair} 20) A) {Siamese}, {domestic shorthair}, { } B) {Siamese, domestic shorthair}, {Siamese}, {domestic shorthair}, { } C) {Siamese, domestic shorthair}, {Siamese}, {domestic shorthair}, D) {Siamese, domestic shorthair}, {domestic shorthair}, { } Write or in the blank so that the resulting statement is true. 21) {red, blue, green} {blue, green, yellow, black} 21) A) B) Calculate the number of subsets and the number of proper subsets for the set. 22) the set of natural numbers less than 10 22) A) 511; 510 B) 510; 511 C) 512; 511 D) 511; 512 Consider below the branching tree diagram based on the number per 3000 American adults. Let T = the set of Americans who like classical music R = the set of Republicans who like classical music D = the set of Democrats who like classical music I = the set of Independents who like classical music Determine whether the statement is true or false. 23) 23) Let M = the set of Republican men who like classical music W = the set of Republican women who like classical music If x R, then x M. A) True B) False 3

4 Use the Venn diagram to list the elements of the set in roster form. 24) The set of students who studied Saturday 24) A) {Karen, Charly} B) {Karen, Charly, Sam, Sophia} C) {Sam, Sophia} D) {Karen, Charly, Vijay} Let U = {1, 2, 4, 5, a, b, c, d, e}. Use the roster method to write the complement of the set. 25) Q = {2, 4, b, d} 25) A) {1, 3, 5, a, c, e} B) {1, 5, a, c, e} C) {1, 5, a, e} D) {1, 2, 4, 5, a, b, c, d, e} Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z}. List the elements in the set. 26) (A C)' 26) A) {w, y} B) {q, r, s, t, u, v, x, z} C) {q, r, s, t, u, v, w, x, y, z} D) {q, s, y, z} 27) 27) C A) {q, s, y, z} B) {v, w, x, y, z} C) { } D) {q, s, u, w, y} Use the Venn diagram to list the elements of the set in roster form. 28) 28) (A B)' A) {13, 17} B) {18, 19} C) {11, 12, 14, 15, 16} D) {11, 12, 13, 14, 15, 16, 17} 4

5 Use sets to solve the problem. 29) Results of a survey of fifty students indicate that 30 like red jelly beans, 29 like green jelly beans, 29) and 17 like both red and green jelly beans. How many of the students surveyed like red or green jelly beans? A) 25 B) 13 C) 42 D) 17 Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z}. List the elements in the set. 30) A' B 30) A) {s, u, w} B) {q, s, t, u, v, w, x, y} C) {r, s, t, u, v, w, x, z} D) {q, r, s, t, v, x, y, z} 31) 31) (A B) (A C) A) {q, s, u, w, y, z} B) {r, t, v, x} C) {q, s, u, w, y} D) {q, s, w, y} 32) 32) (A' B) (A' C') A) {q, s, u, v, x, y} B) {r, s, t, y, z} C) {q, r, t, y, z} D) {r, t, z} 33) (A B C)' 33) A) B) {r, t, v, x} C) {q, r, s, t, u, v, w, x, z} D) {q, s, u, w, z} Use the Venn diagram shown to answer the question. 34) 34) Which regions represent set E'? A) II, III, V, VI B) I, IV, VII, VIII C) II, V, VI D) VIII 5

6 Determine whether the sentence is a statement. 35) Does she always act like that when she's unhappy? 35) A) not a statement B) statement Form the negation of the statement. 36) Copenhagen is not the capital of Brazil. 36) A) It is not true that Copenhagen is not the capital of Brazil. B) It is not true that Brazil is not the capital of Copenhagen. C) It is true that Brazil is not the capital of Copenhagen. D) It is true that Copenhagen is not the capital of Brazil. Let p, q, r, and s represent the following statements: p: One plays hard. q: One is a guitar player. r: The commute to work is not long. s: It is not true that the car is working. Express the following statement symbolically. 37) 37) The car is working. A) ~s B) s C) ~r D) r Express the symbolic statement ~p in words. 38) p: Lake Champlain is one of the Great Lakes. 38) A) It is true that Lake Champlain is one of the Great Lakes. B) Lake Champlain is not one of the Great Lakes. C) Lake Champlain is truly one of the Great Lakes. D) It is not true that Lake Champlain is not one of the Great Lakes Express the quantified statement in an equivalent way, that is, in a way that has exactly the same meaning. 39) Some mammals are cats. 39) A) At least one mammal is a cat. B) There exists at least one cat that is a mammal. C) No mammals are cats. D) All cats are mammals. 40) Some drinks are not liquids. 40) A) All drinks are liquids. B) Some liquids are not drinks. C) All drinks are not liquids. D) Not all drinks are liquids. Write the negation of the quantified statement. (The negation should begin with "all," "some," or "no.") 41) Some drinks are not liquids. 41) A) All drinks are liquids. B) All liquids are drinks. C) All drinks are not liquids. D) No drinks are liquids. 6

7 Construct a truth table for the statement. 42) (r q) (~r ~q) 42) A) r q (r q) (~r ~q) B) r q (r q) (~r ~q) T F F F T F C) r q (r q) (~r ~q) T T T T F T F T T F F F T T F T F F F T T F F T D) r q (r q) (~r ~q) T T T T F F F T F F F T 43) 43) ~(r t) ~(t r) A) r t ~(r t) ~(t r) T T F T F F F T F F F F C) r t ~(r t) ~(t r) T T F T F F F T F F F T B) r t ~(r t) ~(t r) T T F T F T F T T F F F D) r t ~(r t) ~(t r) T T F T F F F T T F F F 44) (p q) (p q) 44) A) B) p q p q p q (p q) (p q) T T T T T T F T F F F T T F F F F F F T p q p q p q (p q) (p q) T T T T T T F F T T F T F T T F F F F T C) p q p q p q (p q) (p q) T T T T T T F F T T F T F T T F F T F F D) p q p q p q (p q) (p q) T T T T T T F F T T F T F T T F F F F F 7

8 45) 45) (q ~r) (q ~r) A) q r (q ~r) (q ~r) T T T T F T F T F F F T C) q r (q ~r) (q ~r) T T F T F F F F F F T T B) q r (q ~r) (q ~r) T T F T F T F T T F F T D) q r (q ~r) (q ~r) T T T T F T F T F F F F Construct a truth table for the given statement and then determine if the statement is a tautology. 46) [ (p ~ q) q ] p 46) A) p q ~ q p ~ q (p ~ q) q [ (p ~ q) q ] p T T F F F T T F T T T T Is not a tautology. F T F F T F F F T T T F B) C) D) p q ~ q p ~ q (p ~ q) q [ (p ~ q) q ] p T T F F T T T F T T T T F T F T T F F F T T T F p q ~ q p ~ q (p ~ q) q [ (p ~ q) q ] p T T F F T T T F T T T T F T F T T F F F T T T F p q ~ q p ~ q (p ~ q) q [ (p ~ q) q ] p T T F F T T T F T T T T F T F T T T F F T T T T Is a tautology. Is not a tautology. Is a tautology. 8

9 47) [ (p ~ q) ~ p ] ~ q 47) A) B) C) D) p q ~ q p ~ q ~ p (p ~ q) ~ p [ (p ~ q) ~ p ] ~ q T T F F F F T T F T T F F T Is a tautology. F T F F T F T F F T F T F T p q ~ q p ~ q ~ p (p ~ q) ~ p [ (p ~ q) ~ p ] ~ q T T F F F F T T F T T F F T Is not a tautology. F T F F T F T F F T F T F T p q ~ q p ~ q ~ p (p ~ q) ~ p [ (p ~ q) ~ p ] ~ q T T F F F F T T F T T F F F Is not a tautology. F T F F T F T F F T F T F F p q ~ q p ~ q ~ p (p ~ q) ~ p [ (p ~ q) ~ p ] ~ q T T F F F F F T F T T F F F Is not a tautology. F T F F T F F F F T F T F F 9

10 48) 48) ( ~ p q) (~ p q) A) p q ~ p ~ p q ~ p q ( ~ p q) (~ p q) B) C) D) T T F F T T T F F F F F F T T T T T F F T F T T Is not a tautology. p q ~ p ~ p q ~ p q ( ~ p q) (~ p q) T T F F F F T F F F F F Is not a tautology. F T T T T T F F T F F F p q ~ p ~ p q ~ p q ( ~ p q) (~ p q) T T F F T F T F F F F F Is not a tautology. F T T T T T F F T F T F p q ~ p ~ p q ~ p q ( ~ p q) (~ p q) T T F F T T T F F F F T Is a tautology. F T T T T T F F T F T T 10

11 Use a truth table to show that p q and ~ p q are equivalent. Then use the result to write a statement that is equivalent to the statement shown. 49) If the garden is not watered every day, the flowers wilt. 49) A) p q p q ~ p ~ p q T T T F T T F F F F F T T T T F F T T T B) C) D) The flowers do not wilt if and only if the garden is not watered every day. p q p q ~ p ~ p q T T T F T T F F F F F T T T T F F T T T Either the garden is not watered every day, or the flowers don't wilt. p q p q ~ p ~ p q T T T F T T F F F F F T T T T F F T T T Either the garden is watered every day, or the flowers wilt. p q p q ~ p ~ p q T T T F T T F F F F F T T T T F F T T T The flowers wilt if the garden is not watered every day. Write the negation of the conditional statement. 50) If she can't water the lawn, I will. 50) A) She can't water the lawn, I can't. B) She can water the lawn, and I can't. C) If she can water the lawn, I can't. D) She can't water the lawn, and I won't. 11

12 Use the De Morgan law that states: ~(p q) is equivalent to ~ p ~ q to write an equivalent English statement for the statement. 51) 51) It is not true that condors and rabbits are both birds. A) condors are birds or rabbits are birds. B) Neither condors nor rabbits are birds. C) rabbits are not birds, but condors are. D) condors are not birds or rabbits are not birds. Use a truth table to determine whether the symbolic form of the argument is valid or invalid. 52) (p q) (q p) 52) q p q A) B) C) D) p q p q q p [ (p q) (q p) ] p q [ [(p q) (q p) ] q ] (p q) T T T T T F T T F F T F F T F T T F F F T F F T T T T F Argument is invalid. p q p q q p [ (p q) (q p) ] p q [ [( p q) (q p) ] q ] (p q) T T T T T T T T F F T F T T F T T F F T T F F T T T F T Argument is valid. p q p q q p [ (p q) (q p) ] p q [ [( p q) (q p) ] q ] (p q) T T T T T T T T F F T F F T F T T F F F T F F T T T F F Argument is invalid. p q p q q p [ (p q) (q p) ] p q [ [(p q) (q p) ] q ] (p q) T T T T T T T T F F T F T T F T T F F T T F F T T T F F Argument is valid. 12

13 53) 53) q p p r r q A) B) C) p q r q p p r (q p) (p r) r q [(q p) (p r)] (r q) T T T T T T T T T T F T F F T F T F T T T T F F T F F T F F T F F T T F T F T F F T F F T F T F F F T T T T F F F F F T T T T T Symbolic argument is invalid. p q r q p p r (q p) (p r) r q [(q p) (p r)] (r q) T T T T T T T T T T F T F F T T T F T T T T F F T F F T F F T T F T T F T F T T F T F F T F T T F F T T T T F F F F F T T T T T Symbolic argument is invalid. p q r q p p r (q p) (p r) r q [(q p) (p r)] (r q) T T T T T T T T T T F T F F T T T F T T T F T T T F F T F F T T F T T F T F T T F T F F T F T T F F T T T F T T F F F T T T T T Symbolic argument is valid. 13

14 D) p q r q p p r (q p) (p r) r q [(q p) (p r)] (r q) T T T T T T T T T T F T F F T T T F T T T F T T T F F T F F T T F T T F T F T T F T F F T F T T F F T T T F T T F F F T T F T T Symbolic argument is valid. 54) 54) ~ p q q r p r A) B) p q r ~ p ~ p q q r (~ p q) (q r) p r [(~ p q) (q r)] (p r) T T T F F T F T T T T F F F F F F T T F T F F F F T T T F F F F T F F T F T T T T T T F F F T F T T F F F T F F T T F F F F T F F F T F T F F T Symbolic argument is invalid. p q r ~ p ~ p q q r (~ p q) (q r) p r [(~ p q) (q r)] (p r) T T T F F T F T F T T F F F F F F F T F T F F F F T F T F F F F T F F F F T T T T T T F F F T F T T F F F F F F T T F F F F F F F F T F T F F F Symbolic argument is invalid. 14

15 C) D) p q r ~ p ~ p q q r (~ p q) (q r) p r [(~ p q) (q r)] (p r) T T T F F T F T T T T F F F F F F T T F T F F F F T T T F F F F T F F T F T T T T T T T T F T F T T F F F T F F T T F F F F T F F F T F T F T T Symbolic argument is valid. p q r ~ p ~ p q q r (~ p q) (q r) p r [(~ p q) (q r)] (p r) T T T F F T F T T T T F F F F F F T T F T F F F F T T T F F F F T F F T F T T T T T T T T F T F T T F F F T F F T T F F F F T F F F T F T F F T Symbolic argument is valid. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Translate the argument into symbolic form. Then use a truth table to determine whether the argument is valid or invalid. (Ignore differences in past, present, and future tense.) 55) 55) If Emilio and Rodrigo both cook, then the meal is tasty. Emilio cooked and the meal was not tasty. Rodrigo did not cook. 15

16 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Construct a truth table for the statement. 56) ~(p q) ~(p q) 56) A) p q ~(p q) ~(p q) B) p q ~(p q) ~(p q) T T F T F F F T F F F T C) p q ~(p q) ~(p q) T T T T F F F T F F F T T T T T F T F T T F F T D) p q ~(p q) ~(p q) T T F T F T F T T F F T Write the compound statement in symbolic form. Let letters assigned to the simple statements represent English sentences that are not negated. Use the dominance of connectives to show grouping symbols (parentheses) in symbolic statements. 57) If I do not like the song and I change the station then the DJ is not entertaining or I look for a CD to 57) play. A) (p ~r) (q ~s) B) [~p (r ~q)] s C) (~p r) (~q s) D) ~p [(r ~q) s] Let p, q, and r represent the following simple statements: p: There is a blizzard outside. q: We do not have to go to school. r: We go sledding. First place parenthesis as needed before and after the most dominant connective and then translate the symbolic statement into English. 58) 58) ~r ~p ~q A) If we go sledding, then there is a blizzard outside or we do not have to go to school. B) If we go sledding then there is a blizzard outside, and we do not have to go to school. C) If we do not go sledding then there is not a blizzard outside, or we have to go to school. D) If we do not go sledding, then there is not a blizzard outside or we have to go to school. Given that p and q each represents a simple statement, write the indicated symbolic statement in words. 59) p: The car has been repaired. 59) q: The kids are home. r: We will visit Aunt Tillie. ~ r (~ p ~ q) A) We will not visit Aunt Tillie if and only if the car has not been repaired or the kids are not home. B) If we will not visit Aunt Tillie, then the car has not been repaired or the kids are not home. C) If we visit Aunt Tillie, then the car has been repaired or the kids are home. D) If we will not visit Aunt Tillie, then the car has not been repaired and the kids are not home. 16

17 60) 60) p: Emilio dislikes Laura q: Laura dislikes Emilio ~ (p q) A) Emilio does not dislike Laura, but Laura dislikes Emilio. B) Emilio dislikes Laura and Laura dislikes Emilio. C) Emilio dislikes Laura but Laura does not dislike Emilio. D) It is not true that Emilio dislikes Laura and Laura dislikes Emilio. Given that p and q each represents a simple statement, write the indicated compound statement in its symbolic form. 61) p: The outside humidity is high. 61) q: The basement dehumidifier is running. r: The basement is getting moldy. If the outside humidity is high, then the basement dehumidifier is running or the basement is not getting moldy. A) p (q ~ r) B) p (q r) C) p (q ~ r) D) p (q ~ r) 62) 62) p: The outside humidity is high. q: The basement dehumidifier is running. r: The basement is getting moldy. The outside humidity is high and the basement dehumidifier is running, or the basement is getting moldy. A) p q r B) (p q) ~ r C) p (q r) D) (p q) r 63) 63) p: He works out. q: He builds up his strength. He works out or he does not build up his strength. A) p ~ q B) p ~ q C) p ~ q D) p q Express the symbolic statement ~p in words. 64) p: No fifth graders play soccer. 64) A) All fifth graders play soccer. B) No fifth grader does not play soccer. C) At least one fifth grader plays soccer. D) Not all fifth graders play soccer. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Translate the argument into symbolic form. Then use a truth table to determine whether the argument is valid or invalid. (Ignore differences in past, present, and future tense.) 65) 65) If it is July or August, then I am living at the beach I am not living at the beach. It is neither July nor August. 17

18 Answer Key Testname: 1332 REVIEW FOR EXAM 1 1) C 2) A 3) B 4) A 5) B 6) B 7) B 8) D 9) D 10) B 11) A 12) B 13) A 14) A 15) B 16) B 17) A 18) D 19) A 20) B 21) B 22) C 23) B 24) B 25) B 26) B 27) B 28) B 29) C 30) D 31) A 32) D 33) C 34) B 35) A 36) A 37) A 38) B 39) A 40) D 41) A 42) D 18

19 Answer Key Testname: 1332 REVIEW FOR EXAM 1 43) C 44) B 45) D 46) C 47) A 48) C 49) C 50) D 51) D 52) B 53) B 54) A 55) p: Emilio cooks. q: Rodrigo cooks. r: The meal is tasty. 56) C 57) C 58) D 59) B 60) D 61) A 62) D 63) A 64) C (p q) r p ~ r ~ q p q r p q (p q) r ~r p ~r [(p q) r] (p ~r) ~ q {[(p q) r ] (p ~r)} ~q T T T T T F F F F T T T F T F T T F F T T F T F T F F F T T T F F F T T T T T T F T T F T F F F F T F T F F T T F F F T F F T F T F F F T T F F F F T T F F T T Argument is valid. 19

20 Answer Key Testname: 1332 REVIEW FOR EXAM 1 65) p: It is July. q: It is August. r: I am living at the beach. (p q) r ~ r ~ p ~ q p q r p q (p q) r ~r [(p q) r ] ~ r ~ p ~ q ~ p ~ q { [(p q) r ] ~ r } ( ~ p ~ q ) T T T T T F F F F F T T T F T F T F F F F T T F T T T F F F T F T T F F T F T F F T F T F T T T T F F T F F T F T F T F T F T F F T F F T F T F F T T T T F F F F T T T T T T T Argument is valid. 20