Review Test 1
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether the statement is true or false. 1) 4 ∈ {1, 2, 3, ..., 15} 1) A) True B) False
2) 21 ∉ {1, 2, 3, ..., 40} 2) A) True B) False
3) 19 ∈ {2, 4, 6, ..., 20} 3) A) True B) False
Fill in the blank with either ∈ or ∉ to make the statement true. 4) Saskatchewan _____ the set of states in the United States 4) A) ∈ B) ∉
Find the cardinal number for the set. 5) {27, 29, 31, 33, 35} 5) A) 5 B) 27 C) 4 D) 6
6) Determine the cardinal number of the set {x | x is a letter of the alphabet} 6) A) 23 B) 26 C) 30 D) 25
Are the sets equivalent? 7) A = {23, 25, 27, 29, 31} 7) B = {24, 26, 28, 30, 32} A) Yes B) No
8) A = {13, 14, 14, 15, 15, 15, 16, 16, 16, 16} 8) B = {16, 15, 14, 13} A) Yes B) No
Are the sets equal? 9) {50, 52, 54, 56, 58} = {52, 54, 56, 58} 9) A) Yes B) No
10) A is the set of residents age 26 or older living in the United States 10) B is the set of residents age 26 or older registered to vote in the United States A) Yes B) No
11) A = {18, 19, 19, 20, 20, 20, 21, 21, 21, 21} 11) B = {21, 20, 19, 18} A) Yes B) No
12) A = {14, 15, 16, 17, 18} 12) B = {13, 14, 15, 16, 17} A) Yes B) No
1 Write ⊆ or ⊈ in the blank so that the resulting statement is true. 13) {4, 34, 39} {15, 34, 39, 49} 13) A) ⊆ B) ⊈
14) {c, a, n, d, i, d, a, t, e} _____ {a, c, d, e, i, t, a, n, d} 14) A) ⊆ B) ⊈
Determine whether the statement is true or false. 15) ∅ ⊆ {France, Germany, Switzerland} 15) A) True B) False
Use ⊆, ⊈, ⊂, or both ⊂ and ⊆ to make a true statement. 16) {a, b} {z, a, y, b, x, c} 16) A) ⊂ and ⊆ B) ⊈ C) ⊂ D) ⊆
List all the subsets of the given set. 17) {8} 17) A) {0}, {8}, { }B){ } C) {8} D) {8}, { }
Calculate the number of subsets and the number of proper subsets for the set. 1 1 1 1 18) , , , 18) 6 7 8 9 A) 15; 14 B) 14; 15 C) 16; 15 D) 15; 16
19) {1, 3, 5, 7, 9, 11} 19) A) 64; 63 B) 63; 62 C) 63; 64 D) 62; 63
20) {x | x is a day of the week} 20) A) 127; 126 B) 64; 65 C) 128; 127 D) 128; 129
2 Place the various elements in the proper regions of the Venn diagram. 21) Let U = {8, 9, 10, 11, 12, 13, 14} and A = {8, 9, 12}. Find Aʹ and place the elements in the proper 21) region.
A) Aʹ = {11, 12, 13, 14} B) Aʹ = {8, 9, 10, 11, 12, 13, 14}
C) Aʹ = {8, 9, 12} D) Aʹ = {10, 11, 13, 14}
Use the Venn diagram to list the elements of the set in roster form. 22) List the elements of A. 22)
A) {11, 13, 14, 17} B) {12, 15, 16} C) {11, 12, 13} D) {13, 17}
3 23) List the elements of U. 23)
A) {11, 14} B) {13, 17} C) {12, 15, 16} D) {11, 12, 13, 14, 15, 16, 17, 18, 19}
Let U = {1, 2, 4, 5, a, b, c, d, e}. Use the roster method to write the complement of the set. 24) A = {2, 4, b, d} 24) A) {1, 2, 4, 5, a, b, c, d, e} B) {1, 5, a, c, e} C) {1, 5, a, e} D) {1, 3, 5, a, c, e}
Let U = {21, 22, 23, ..., 40}, A = {21, 22, 23, 24, 25}, B = {26, 27, 28, 29}, C = {21, 23, 25, 27, ..., 39}, and D = {22, 24, 26, 28, ..., 40}. Use the roster method to write the following set. 25) Aʹ 25) A) Aʹ = {21, 22, 23, . . . , 40} B) Aʹ = {27, 29, 31, . . . , 39} C) Aʹ = {26, 28, 30, . . . , 40} D) Aʹ = {26, 27, 28, . . . , 40}
Let U = {21, 22, 23, 24, ...}, A = {21, 22, 23, 24, ..., 40}, B = {21, 22, 23, 24, ..., 50}, C = {22, 24, 26, 28, ...}, and D = {21, 23, 25, 27, ...}. Use the roster method to write the following set. 26) Cʹ 26) A) Cʹ = {21, 22, 23, 24, ...} B) Cʹ = {21, 23, 25, 27, ..., 39} C) Cʹ = {21, 23, 25, 27, ...} D) Cʹ = {22, 24, 26, 28, ...}
Solve the problem. 27) If the universal set is the set of the days of the week and set A is the set of days that begin with the 27) letter T, write Aʹ using the roster method. Describe Aʹ in words. A) Aʹ = {Sunday, Monday, Friday, Saturday}; Aʹ is the days of the week that do not begin with the letter T. B) Aʹ = {Tuesday, Thursday}; Aʹ is the days of the week that begin with the letter T. C) Aʹ = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}; Aʹ is the days of the week that do not begin with the letter T. D) Aʹ = {Sunday, Monday, Wednesday, Friday, Saturday}; Aʹ is the days of the week that do not begin with the letter T.
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. 28) A ∩ Bʹ 28) A) {r, s, t, u, v, w, x, z} B) {t, v, x} C) {u, w} D) {q, s, t, u, v, w, x, y}
4 29) (A ∩ B)ʹ 29) A) {q, s, t, u, v, w, x, y} B) {s, u, w} C) {r, t, u, v, w, x, z} D) {t, v, x}
30) A ∩ B 30) A) {q, s, y} B) {q, s, u, w, y, z} C) {v, w, x, y, z} D) {r, t, u, v, w, x, z}
31) (A ∪ B)ʹ 31) A) {s, u, w} B) {r, s, t, u, v, w, x, z} C) {r, t, v, x} D) {t, v, x}
32) B ∪ C 32) A) {q, r, s, t, u, v, w, x, y, z} B) {q, s, u, w, y} C) {v, w, x, y, z} D) {q, s, v, w, x, y, z}
33) B ∪ U 33) A) {q, s, u, w, y} B) {q, s, y, z} C) {v, w, x, y, z} D) {q, r, s, t, u, v, w, x, y, z}
34) A ∩ Bʹ 34) A) {r, s, t, u, v, w, x, z} B) {u, w} C) {q, s, t, u, v, w, x, y} D) {t, v, x}
35) Cʹ ∪ Aʹ 35) A) {w, y} B) {q, r, s, t, u, v, x, z} C) {s, t} D) {q, s, u, v, w, x, y, z}
36) B ∪ C 36) A) {q, s, v, w, x, y, z} B) {v, w, x, y, z} C) {q, r, s, t, u, v, w, x, y, z} D) {q, s, u, w, y}
37) (A ∩ C)ʹ 37) 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}
38) C ∪ ∅ 38) A) { } B) {q, s, y, z} C) {v, w, x, y, z} D) {q, s, u, w, y}
39) B ∪ U 39) A) {v, w, x, y, z} B) {q, r, s, t, u, v, w, x, y, z} C) {q, s, u, w, y} D) {q, s, y, z}
5 Use the Venn diagram to list the elements of the set in roster form. 40) 40)
A ∪ B A) {11, 12, 14, 15, 16} B) {13, 17} C) {11, 12, 13, 14, 15, 16, 17, 18, 19} D) {11, 12, 13, 14, 15, 16, 17}
41) 41)
Aʹ A) {12, 15, 16, 18, 19} B) {11, 13, 14, 17} C) {12, 15, 16} D) {11, 14, 18, 19}
42) 42)
(A ∩ B)ʹ A) {18, 19} B) {11, 12, 14, 15, 16} C) {11, 12, 14, 15, 16, 18, 19} D) {13, 17}
6 43) 43)
(A ∪ B)ʹ A) {11, 12, 14, 15, 16} B) {11, 12, 13, 14, 15, 16, 17} C) {18, 19} D) {13, 17}
Use sets to solve the problem. 44) Results of a survey of fifty students indicate that 30 like red jelly beans, 29 like green jelly beans, 44) and 17 like both red and green jelly beans. How many of the students surveyed like no green jelly beans? A) 38 B) 17 C) 30 D) 21
45) Monticello residents were surveyed concerning their preferences for candidates Moore and Allen 45) in an upcoming election. Of the 800 respondents, 300 support neither Moore nor Allen, 100 support both Moore and Allen, and 250 support only Moore. How many residents support Allen? A) 400 B) 100 C) 250 D) 150
Use the formula for the cardinal number of the union of two sets to solve the problem. 46) Set A contains 35 elements and set B contains 22 elements. If there are 40 elements in (A ∪ B) then 46) how many elements are in (A ∩ B)? A) 13 B) 17 C) 5 D) 8
47) Set A contains 5 elements, set B contains 11 elements, and 3 elements are common to sets A and B. 47) How many elements are in A ∪ B? A) 12 B) 14 C) 16 D) 13
48) Set A contains 35 elements and set B contains 22 elements. If there are 40 elements in (A ∪ B) then 48) how many elements are in (A ∩ B)? A) 8 B) 17 C) 5 D) 13
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. 49) A ∪ (B ∩ C) 49) A) {q, w, y} B) {q, r, w, y, z} C) {q, s, u, w, y, z} D) {q, y, z}
50) (A ∩ B) ∪ (A ∩ C) 50) A) {q, s, v, w, y} B) {q, s, u, w, y} C) {r, t, u, v, x, z} D) {q, s, w, y}
51) (A ∪ B ∪ C)ʹ 51) A) {q, s} B) {v, z} C) {r, t} D) {s, t}
7 52) (B ∪ C)ʹ ∩ A 52) A) {w} B) {v} C) ∅ D) {u}
Use the following information to construct a Venn Diagram that illustrates the given sets. 53) U = the set of members of the bookclub shown in the chart 53) A = the set of members of the bookclub who read at least 25 books B = the set of members of the bookclub who suggested 5 or less books C = the set of members of the bookclub who started their membership after 2000
Members of Numbers of books Numbers of books Year of the bookclub read suggested membership Carla 24 7 2001 Marge 25 2 2000 Sandy 5 1 2004 Laura 45 15 1998 Kim 42 11 1998 Peter 29 9 1999 Jim 44 5 2000 Ann 24 1 1998 Paul 17 1 2002
A)
B)
8 C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use the Venn diagram shown below to solve the problem.
54) a) Which regions are represented by (Aʹ ∩ B)ʹ? 54) b) Which regions are represented by A ∩ Bʹ? c) Based on parts a) and b), what can you conclude about the relationship between (Aʹ ∩ B)ʹ and A ∩ Bʹ?
9 55) a) Which regions are represented by (A ∪ Bʹ)ʹ? 55) b) Which regions are represented by Aʹ ∩ B? c) Based on parts a) and b), what can you conclude about the relationship between (A ∪ Bʹ)ʹ and Aʹ ∩ B?
Use the Venn diagram shown below to solve the problem.
56) a) Which regions are represented by (A ∩ B) ∪ C? 56) b) Which regions are represented by (A ∪ C) ∩ (A ∪ B)? c) Based on parts a) and b), what can you conclude about the relationship between (A ∩ B) ∪ C and (A ∪ C) ∩ (A ∪ B)?
57) a) Which regions are represented by B ∪ (A ∩ C)? 57) b) Which regions are represented by (A ∪ B) ∩ (B ∪ C)? c) Based on parts a) and b), what can you conclude about the relationship between B ∪ (A ∩ C) and (A ∪ B) ∩ (B ∪ C)?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the accompanying Venn diagram that shows the number of elements in regions I through IV to answer the question. 58) 58)
6 13 12
7
How many elements belong to set B? A) 12 B) 19 C) 25 D) 18
10 59) 59)
9 15 14
10
How many elements belong to set A or set B? A) 48 B) 29 C) 9 D) 38
Use the given cardinalities to determine the number of elements in the specific region. 60) n(U) = 147, n(A) = 48, n(B) = 68, n(C) = 46, n(A ∩ B) = 19, n(A ∩ C) = 22, n(B ∩ C) = 18, n(A ∩ B ∩ C) 60) = 10 Find III.
A) 41 B) 14 C) 30 D) 26
11 61) n(U) = 189, n(A) = 126, n(B) = 94, n(C) = 92, n(A ∩ B) = 52, , n(A ∩ C) = 49, n(B ∩ C) = 47, n(A ∩ B ∩ 61) C) = 25 Find VI.
A) 22 B) 23 C) 21 D) 24
Use a Venn diagram to answer the question. 62) At East Zone University (EZU) there are 565 students taking College Algebra or Calculus. 327 are 62) taking College Algebra, 266 are taking Calculus, and 28 are taking both College Algebra and Calculus. How many are taking Calculus but not Algebra? A) 537 B) 299 C) 238 D) 271
63) A local television station sends out questionnaires to determine if viewers would rather see a 63) documentary, an interview show, or reruns of a game show. There were 400 responses with the following results: 120 were interested in an interview show and a documentary, but not reruns; 16 were interested in an interview show and reruns, but not a documentary; 56 were interested in reruns but not documentaries or interviews; 96 were interested in an interview show but not a documentary; 40 were interested in a documentary and reruns; 24 were interested in an interview show and reruns; 32 were interested in none of the three. How many are interested in exactly one kind of show? A) 172 B) 192 C) 182 D) 202
Determine whether the sentence is a statement. 64) Does Britta always act like that in public? 64) A) not a statement B) statement
65) 5 + 4 = 11 65) A) not a statement B) statement
66) Do you like this color? 66) A) not a statement B) statement
12 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. 67) One does not play hard. 67) A) ~q B) q C) p D) ~p
68) The commute to work is long. 68) A) s B) ~rC)rD)~s
Form the negation of the statement. 69) Today is May 10 69) A) Today is not May 10. B) It is not true that today is May 11. C) Today is not May 11. D) Yesterday was not May 8.
Express the symbolic statement ~p in words. 70) p: The Pilgrims did not land in Tahiti. 70) A) The Pilgrims landed in Tahiti. B) The Pilgrims landed on Plymouth Rock. C) The Pilgrims almost landed in Tahiti. D) It is not true that the Pilgrims landed in Tahiti.
71) p: Vitamin C helps the immune system. 71) A) Vitamin C may help the immune system. B) Vitamin C does not help the immune system. C) It is true that Vitamin C helps the immune system. D) Vitamin A helps the immune system.
72) p: The refrigerator is not working. 72) A) The oven is working. B) The refrigerator is working. C) It is not true that the refrigerator is working. D) The refrigerator is almost working.
Express the quantified statement in an equivalent way, that is, in a way that has exactly the same meaning. 73) All uncles are males. 73) A) There are no uncles that are not males. B) Some males are not uncles. C) All males are not uncles. D) At least one uncle is a male.
74) Some flowers are roses. 74) A) No flowers are roses. B) At least one flower is a rose. C) There exists at least one rose that is a flower. D) All roses are flowers.
75) Some violinists are not humans. 75) A) Not all violinists are humans. B) Some humans are not violinists. C) All violinists are not humans. D) All violinists are humans.
13 Write the negation of the quantified statement. (The negation should begin with ʺall,ʺ ʺsome,ʺ or ʺno.ʺ) 76) Some mammals are horses. 76) A) All horses are mammals. B) No horses are mammals. C) No mammals are horses. D) Not all mammals are horses.
77) No South American soccer teams have won a World Cup. 77) A) Some South American soccer teams have not won a World Cup. B) Some South American soccer teams have won a World Cup. C) All South American soccer teams have not won a World Cup. D) All South American soccer teams have won a World Cup.
78) All athletes are famous. 78) A) All athletes are somewhat famous. B) Some athletes are famous. C) Some athletes are not famous. D) All athletes are not famous.
Choose the correct conclusion. 79) As a special promotion, the Green Thumb organic foods chain said that everyone who came to one 79) of their stores between noon and 1 p.m. on January 4 would be offered a free loaf of 23 -grain bread. They did not keep this promise. Therefore, between noon and 1 p.m. on January 4: A) At least one person who came to a Green Thumb store was not offered a free loaf of 23 -grain bread. B) The Green Thumb chain ran out of 23-grain bread. C) At least one person who came to a Green Thumb store was offered a free loaf of 23-grain bread. D) No one who came to a Green Thumb store was offered a free loaf of 23 -grain bread.
Express the symbolic statement ~p in words. 80) p: Not all people like football. 80) A) All people like football. B) All people do not like football. C) Some people like football. D) Some people do not like football.
81) p: Some athletes are musicians. 81) A) Some athletes are not musicians. B) No athlete is a musician. C) Not all athletes are musicians. D) All athletes are musicians.
82) p: Some people donʹt like walking. 82) A) Nobody likes walking. B) Some people like walking. C) Some people donʹt like driving. D) Everyone likes walking.
Given that p and q each represents a simple statement, write the indicated compound statement in its symbolic form. 83) p: Spartacus is a film. 83) q: Rambo is a film. Spartacus is a film and Rambo is a film. A) p ∧ ~ qB)p → qC)p ∨ qD)p ∧ q
84) p: They set the alarm. 84) q: They get up on time. They set the alarm or they get up on time. A) p ∨ qB)p ∨ ~ qC)p → qD)p ∧ q
14 85) p: This is a brontosaurus. 85) q: This is a dinosaur. If this is not a brontosaurus, then this is not a dinosaur. A) ~ q → ~ pB)~ p → ~ qC)~ p ∧ ~ qD) p → ~ q
86) p: The outside humidity is high. 86) 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)
87) p: The outside humidity is low. 87) q: The central humidifier is running. r: The air in the house is getting dry.
It is not the case that if the air in the house is getting dry, then the central humidifier is not running. A) ~ (r → q) B) ~ (r → ~ q) C) ~ r → ~ qD)~ (r ∧ ~ q)
88) p: The outside humidity is high. 88) q: The basement dehumidifier is running. r: The basement is getting moldy.
If the basement dehumidifier is running, then the outside humidity is high if and only if the basement is getting moldy. A) q → (p ∧ r) B) p → (q ↔ r) C) (q → p) ↔ rD)q → (p ↔ r)
Given that p and q each represents a simple statement, write the indicated symbolic statement in words. 89) p: Roger likes Vanessa 89) q: Vanessa likes Roger ~ (p ∧ q) A) Roger does not like Vanessa, but Vanessa likes Roger. B) Roger likes Vanessa and Vanessa likes Roger. C) Roger likes Vanessa but Vanessa does not like Roger. D) It is not true that Roger likes Vanessa and Vanessa likes Roger.
90) p: Darren dislikes Zoe 90) q: Zoe dislikes Darren ~ q ∨ p A) Zoe and Darren do not dislike each other. B) It is not true that Zoe dislikes Darren or that Darren dislikes Zoe. C) Zoe does not dislike Darren, but Darren dislikes Zoe. D) Zoe does not dislike Darren, or Darren dislikes Zoe.
15 91) p: The car has been repaired. 91) q: The kids are home. r: We will visit Aunt Tillie. (p ∧ q) → r A) We will visit Aunt Tillie if and only if the car has been repaired and the kids are home. B) If the car has been repaired or the kids are home, we will visit Aunt Tillie. C) If the car has been repaired and the kids are home, we will visit Aunt Tillie. D) If the car has been repaired, we will visit Aunt Tillie even if the kids are not home.
92) p: The fan is working. 92) q: The bedroom is stuffy. p ∨ ~ q A) If the fan is not working, then the bedroom is stuffy. B) The fan is not working or the bedroom is stuffy. C) If the fan is working, then the bedroom is not stuffy. D) The fan is working or the bedroom is not stuffy.
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. 93) If I like the song or the DJ is entertaining then I do not change the station. 93) A) (p ∨ q) → rB)p ∨ (q → r) C) p ∨ (q → ~r) D) (p ∨ q) → ~r
94) 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 94) play. A) ~p ∧ [(r → ~q) ∨ s] B) (p ∧ ~r) → (q ∨ ~s) C) (~p ∧ r) → (~q ∨ s) D) [~p ∧ (r → ~q)] ∨ s
Write the statement in symbolic form to determine the truth value for the statement. 95) Miami is a city and China is a country. 95) A) True B) False
96) 5 × 2 = 10 or French is a language. 96) A) True B) False
16 Complete the truth table by filling in the required columns. 97) p ∧ ~ q 97)
p q ~ q p ∧ ~ q T T T F F T F F A) B) p q ~ q p ∧ ~ q p q ~ q p ∧ ~ q T T F F T T F T T F T T T F T T F T F F F T F F F F T T F F T F
C) D) p q ~ q p ∧ ~ q p q ~ q p ∧ ~ q T T F F T T F F T F T T T F T T F T T F F T F F F F T F F F T F
98) ~ (p ∧ q) 98)
p q p ∧ q ~ (p ∧ q) T T T F F T F F A) B) p q p ∧ q ~ (p ∧ q) p q p ∧ q ~ (p ∧ q) T T T F T T T T T F F T T F F F F T F T F T F F F F F F F F F F
C) D) p q p ∧ q ~ (p ∧ q) p q p ∧ q ~ (p ∧ q) T T T F T T T F T F F T T F F T F T F T F T F T F F T F F F F T
17 Construct a truth table for the statement. 99) ~(q ∨ t) ∧ ~(t ∧ q) 99) A) q t ~(q ∨ t) ∧ ~(t ∧ q) B) q t ~(q ∨ t) ∧ ~(t ∧ q) TT F TT F TF F TF F FT T FT F FF F FF F C) q t ~(q ∨ t) ∧ ~(t ∧ q) D) q t ~(q ∨ t) ∧ ~(t ∧ q) TT F TT F TF F TF T FT F FT T FF T FF F
100) ~(p ∨ q) ∧ ~ p 100) A) B) p qp ∨ q ~(p ∨ q) ~ p ~(p ∨ q) ∧ ~ p p qp ∨ q ~(p ∨ q) ~ p ~(p ∨ q) ∧ ~ p T T T F F F T T T F F F T F T F F F T F T F F F F T T F T F F T T T T T F F F T T T F F F T T T
C) D) p qp ∧ q ~(p ∨ q) ~ p ~(p ∧ q) ∧ ~ p p qp ∨ q ~(p ∨ q) ~ p ~(p ∨ q) ∧ ~ p T T T F F F T T T F F F T F F T F F T F T F F F F T F T T T F T T F T T F F T F T F F F F T T T
101) (p ∧ ~t) ∧ q 101) A) p t q (p ∧ ~t) ∧ q B) p t q (p ∧ ~t) ∧ q TTT F TTT F TTF F TTF F TFT T TFT F TFF F TFF F FTT F FTT F FTF F FTF T FFT F FFT T FFF F FFF T C) p t q (p ∧ ~t) ∧ q D) p t q (p ∧ ~t) ∧ q TTT T TTT F TTF T TTF F TFT T TFT F TFF F TFF F FTT F FTT F FTF T FTF T FFT T FFT T FFF T FFF F
18 Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement. 102) ~p ∨ (q ∧ ~r) 102) A) True B) False
103) (p ∧ ~q) ∧ r 103) A) True B) False
104) ~(~p ∧ ~q) ∨ (~r ∨ ~p) 104) A) True B) False
Construct a truth table for the statement. 105) ~ p → ~ q 105) A) B) p q ~ p ~ q ~ p → ~ q p q ~ p ~ q ~ p → ~ q T T F F F T T F F T T F F T T T F F T T F T T F F F T T F F F F T T T F F T T T
C) D) p q ~ p ~ q ~ p → ~ q p q ~ p ~ q ~ p → ~ q T T F F T T T F T T T F F T F T F F F T F T T F F F T T T T F F T T T F F T F F
106) ~(q → ~ p) 106) A) B) p q ~ pq → ~ p ~(q → ~ p) p q ~ pq → ~ p ~(q → ~ p) T T F F T T T F F T T F F F T T F F T F F T T T F F T T T T F F T T F F F T T F
C) D) p q ~ pq → ~ p ~(q → ~ p) p q ~ pq → ~ p ~(q → ~ p) T T F T F T T F F T T F F F T T F F T F F T T F T F T T T F F F T F T F F T T F
19 Construct a truth table for the given statement and then determine if the statement is a tautology. 107) [ (p → ~ q) ∧ q ] → ~ p 107) A) p q ~ q p → ~ q (p → ~ q) ∧ q ~ p [ (p → ~ q) ∧ q ] → ~ p T T F F F F T T F T T T F T Is a tautology. F T F T F T T F F T T F T T
B) p q ~ q p → ~ q (p → ~ q) ∧ q ~ p [ (p → ~ q) ∧ q ] → ~ p T T F F F F T T F T T F F T Is a tautology. F T F T T T T F F T T F T T
C) p q ~ q p → ~ q (p → ~ q) ∧ q ~ p [ (p → ~ q) ∧ q ] → ~ p T T F F F F F T F T T F F F Is not a tautology. F T F T T T T F F T T F T T
D) p q ~ q p → ~ q (p → ~ q) ∧ q ~ p [ (p → ~ q) ∧ q ] → ~ p T T F T T F T T F T F F F T Is a tautology. F T F F F T T F F T F F T T
20 108) [ (p ∧ ~ q) ∧ ~ p ] → ~ q 108) A) 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
B) 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
C) 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
D) 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
Construct a truth table for the given statement. 109) ~ (p ↔ ~ q) 109) A) B) p q ~ q p ↔ ~ q ~ (p ↔ ~ q) p q ~ q p ↔ ~ q ~ (p ↔ ~ q) T T F F T T T F F T T F T T T T F T T F F T F T T F T F T F F F T F T F F T F T
C) D) p q ~ q p ↔ ~ q ~ (p ↔ ~ q) p q ~ q p ↔ ~ q ~ (p ↔ ~ q) T T F F T T T F T F T F T T F T F T F T F T F T F F T F F T F F T T F F F T T F
21 Construct a truth table for the statement. Then determine if the statement is a tautology. 110) ~(~p ∧ q) ↔ (p ∧ ~ q) 110) A) p q ~p ~p ∧ q ~( ~p ∧ q) ~ q p ∧ ~ q ~( ~p ∧ q) ↔ (p ∧ ~ q) T T F F T F F F T F F F T T T T F T T T F F F T F F T F T T F T Is not a tautology.
B) p q ~p ~p ∧ q ~( ~p ∧ q) ~ q p ∧ ~ q ~( ~p ∧ q) ↔ (p ∧ ~ q) T T F F T F F T T F F F T T T T F T T T F F F T F F T F T T F T Is a tautology.
C) p q ~p ~p ∧ q ~( ~p ∧ q) ~ q p ∧ ~ q ~( ~p ∧ q) ↔ (p ∧ ~ q) T T F F T F F F T F F F T T T T F T T T F F F T F F T F T T F F Is not a tautology.
D) p q ~p ~p ∧ q ~( ~p ∧ q) ~ q p ∧ ~ q ~( ~p ∧ q) ↔ (p ∧ ~ q) T T F F T F T T T F F F T T T T F T T T F F F T F F T F T T T T Is a tautology.
22 111) (q → p) ↔ (~ p ∨ q) 111) A) p q q → p ~ p ~ p ∨ q (q → p) ↔ (~ p ∨ q) T T T T T F T F T F F F Is not a tautology. F T F T T F F F F T T T
B) p q q → p ~ p ~ p ∨ q (q → p) ↔ (~ p ∨ q) T T T F T T T F T F F F Is not a tautology. F T F T T F F F T T T T
C) p q q → p ~ p ~ p ∨ q (q → p) ↔ (~ p ∨ q) T T T F T T T F T F F F Is not a tautology. F T F T T T F F T T T T
D) p q q → p ~ p ~ p ∨ q (q → p) ↔ (~ p ∨ q) T T T F T T T F F F F T Is a tautology. F T T T T T F F T T T T
Write the contrapositive of the statement. 112) If I am in the city of Jokdrifa, then I am on the planet Plochus. 112) A) If I am not on the planet Plochus, then I am in the city of Jokdrifa. B) If I am not in the city of Jokdrifa, then I am not on the planet Plochus. C) If I am not on the planet Plochus, then I am not in the city of Jokdrifa. D) If I am not in the city of Jokdrifa, then I am on the planet Plochus.
113) If the electricity is out, then I cannot use the computer. 113) A) If I can use the computer, then the electricity is not out. B) If the electricity is not out, then I can use the computer. C) If I cannot use the computer, then the electricity is out. D) If the electricity is not out, then I cannot use the computer.
114) If he is not working in China, then he is vacationing in England. 114) A) If he is not vacationing in China, then he is working in England. B) If he is not vacationing in England, then he is working in China. C) If he is vacationing in China, then he is not working in England. D) If he is working in China, then he is not vacationing in England.
23 Write the converse and inverse of the statement. 115) If you drink too much coffee, then you get hyper. 115) A) converse: If you get hyper, then you are drinking too much coffee. inverse: If you donʹt get hyper, then you are not drinking too much coffee. B) converse: If you get hyper, then you are drinking too much coffee. inverse: If you donʹt drink too much coffee, you donʹt get hyper. C) converse: If you get hyper, then you are drinking too much coffee. inverse: If you donʹt get hyper, then you are drinking too much coffee. D) converse: If you donʹt drink too much coffee, you donʹt get hyper. inverse: If you get hyper, then you are drinking too much coffee.
116) If you are getting a haircut, then you are not studying. 116) A) converse: If you are not getting a haircut, then you are studying inverse: If you are not studying, then you are getting a haircut. B) converse: If you are not studying, then you are getting a haircut. inverse: If you are not getting a haircut, then you are studying C) converse: If you are studying, then you are not getting a haircut. inverse: If you are not getting a haircut, then you are studying D) converse: If you are not studying, then you are getting a haircut. inverse: If you are getting a haircut, then you are not studying
Write the negation of the conditional statement. 117) If I am in Milwaukee, then I am in Wisconsin. 117) A) I am not in Milwaukee and I am in Wisconsin. B) If I am in Milwaukee, then I am not in Wisconsin. C) If I am not in Milwaukee, then I am not in Wisconsin. D) I am in Milwaukee and I am not in Wisconsin.
118) If she canʹt take out the trash, I will. 118) A) She canʹt take out the trash, I canʹt. B) She can take out the trash, and I canʹt. C) She canʹt take out the trash, and I wonʹt. D) If she can take out the trash, I canʹt.
119) If I get a high-paying job, then I can pay off all my bills. 119) A) I get a high-paying job and can pay off all my bills. B) I donʹt get a high-paying job and cannot pay off all my bills. C) I get a high-paying job and I cannot pay off all my bills. D) I donʹt get a high-paying job and can pay off all my bills.
Use the De Morgan law that states:
~(p ∧ q) is equivalent to ~ p ∨ ~ q to write an equivalent English statement for the statement. 120) It is not true that condors and rabbits are both birds. 120) A) condors are birds or rabbits are birds. B) rabbits are not birds, but condors are. C) Neither condors nor rabbits are birds. D) condors are not birds or rabbits are not birds.
24 121) It is not true that Ireland and Utah are both cities. 121) A) If Ireland is a city, then Utah is not a city. B) Ireland is not a city or Utah is not a city. C) It is true that Ireland and Utah are both cities. D) Ireland is not a city and Utah is not a city.
Use De Morganʹs laws to write a negation of the statement. 122) Roger or Emil will attend the game. 122) A) Roger will not attend the game and Emil will not attend the game. B) Roger will not attend the game and Emil will attend the game. C) Roger or Emil will not attend the game. D) Roger and Emil will attend the game.
123) A man eats six hot dogs and he does not get a stomach ache. 123) A) A man eats six hot dogs and he gets a stomach ache. B) A man eats six hot dogs or he gets a stomach ache. C) A man does not eat six hot dogs or he gets a stomach ache. D) A man does not eat six hot dogs or he does not get a stomach ache.
Determine which, if any, of the three given statements are equivalent. 124) a. If she does not misplace her keys or forgets her phone, then she will get locked out. 124) b. If she misplaces her keys and does not forget her phone then she will not get locked out. c. If she did not get locked out, then she misplaced her keys or did not forget her phone. A) a and bB)b and cC)a and c D) none
Use a truth table to determine whether the symbolic form of the argument is valid or invalid. 125) p → q 125) q ∴ p A) Valid B) Invalid
126) ~q ∧ ~p 126) p ∨ ~q ∴ ~q A) Valid B) Invalid
127) p → q 127) ~p ∴ ~q A) Valid B) Invalid
128) ~p → q 128) ~q → p ∴ p ∨ q A) Valid B) Invalid
25 129) p → q 129) ~ p ∴ ~ q
A) p qp→q ~ p(p → q) ∧ ~ p ~ q[ (p → q) ∧ ~ p ] → ~ q T T T F F F T T F F F T T T The argument is invalid. F T T T T F F F F T T T T T
B) p qp→q ~ p(p → q) ∧ ~ p ~ q[ (p → q) ∧ ~ p ] → ~ q T T T F F F T T F F F F T T The argument is valid. F T F T F F T F F F T F T T
C) p qp→q ~ p(p → q) ∧ ~ p ~ q[ (p → q) ∧ ~ p ] → ~ q T T T F F F T T F F F F T T The argument is invalid. F T T T T F F F F T T T T T
D) p qp→q ~ p(p → q) ∧ ~ p ~ q[ (p → q) ∧ ~ p ] → ~ q T T T F T F T T F F F F T T The argument is valid. F T F T T F T F F F T T T T
26 130) (p → q) ∧ (q → p) 130) q ∴ p ∨ q
A) p qp → qq → 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.
B) p qp → qq → 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.
C) p qp → qq → 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.
D) p qp → qq → 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.
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.) 131) There must be a cease-fire or there is fighting. 131) There is fighting. ∴ There is no cease-fire.
132) If this is Germany or Austria, then the signs are in German. 132) The signs are in German. ∴ This is Germany or Austria
27 Draw a valid conclusion from the given premises. 133) If a person is a competition swimmer, then that person 133) can swim continously for one hour or more. Sharon cannot swim for more than 30 minutes. Therefore....
134) If all taxi drivers are on strike, there are no taxis picking up passengers. 134) Some taxis are picking up passengers. Therefore....
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
135) If I get robbed, I will go to court. 135) I got robbed. Therefore.... A) I will go to court. B) I will not go to court. C) I will not get robbed in court. D) I will get robbed in court.
136) Every man with a mind can think. A distracted man canʹt think. A man who is not distracted can 136) apply himself. Therefore.... A) Every man with a mind is distracted. B) Every man with a mind can apply himself. C) Every man who can apply himself has a mind. D) Every distracted man can apply himself.
Use an Euler diagram to determine whether the argument is valid or invalid. 137) All doctors have studied chemistry. 137) All surgeons are doctors. Therefore, all surgeons have studied chemistry. A) valid B) invalid
138) All dogs like food. 138) All pets like food. Therefore, all dogs are pets. A) valid B) invalid
139) All insects have six legs. 139) No spiders are insects. Therefore, no spiders have six legs. A) valid B) invalid
140) No cheap CDs sound good. 140) Some cheap CDs have red tags. Therefore, some CDs with red tags do not sound good. A) valid B) invalid
28 Answer Key Testname: REVIEW PRESTAT TEST 1
1) A 2) B 3) B 4) B 5) A 6) B 7) A 8) B 9) B 10) B 11) A 12) B 13) B 14) A 15) A 16) A 17) D 18) C 19) A 20) C 21) D 22) A 23) D 24) B 25) D 26) C 27) D 28) C 29) C 30) A 31) C 32) D 33) D 34) B 35) B 36) A 37) B 38) C 39) B 40) D 41) A 42) C 43) C 44) D 45) C 46) B 47) D 48) B 49) C 50) D 29 Answer Key Testname: REVIEW PRESTAT TEST 1
51) C 52) D 53) D 54) a) I, II, and IV b) Ic) They are not equal. 55) a) III b) III c) They are equal. 56) a) II, IV, V, VI, and VII b) I, II, IV, V, and VI c) They are not equal. 57) a) II - VI b) II - VI c) They are equal. 58) D 59) D 60) A 61) A 62) C 63) B 64) A 65) B 66) A 67) D 68) B 69) A 70) A 71) B 72) B 73) A 74) B 75) A 76) C 77) B 78) C 79) A 80) A 81) B 82) D 83) D 84) A 85) B 86) D 87) B 88) D 89) D 90) D 91) C 92) D 93) D 94) C 95) A 96) A 97) D 98) D 99) C 100) A 30 Answer Key Testname: REVIEW PRESTAT TEST 1
101) A 102) B 103) B 104) A 105) B 106) D 107) B 108) B 109) B 110) C 111) B 112) C 113) A 114) B 115) B 116) B 117) D 118) C 119) C 120) D 121) B 122) A 123) C 124) D 125) B 126) A 127) B 128) A 129) C 130) D 131) p: There is a cease-fire. q: There is fighting.
p ∨ q q ∴ ~ p
p qp ∨ q(p ∨ q) ∧ q ~ p[ (p ∨ q) ∧ q ] → ~ p T T T T F F T F T F F T Argument is invalid. F T T T T T F F F F T T
31 Answer Key Testname: REVIEW PRESTAT TEST 1
132) p: This is Germany. q: This is Austria. r: The signs are in German.
(p ∨ q) → r r ∴ p ∨ q
p q rp ∨ q(p ∨ q) → r[ (p ∨ q) → r ] ∧ r{ [ (p ∨ q) → r ] ∧ r } → (p ∨ q) T T T T T T T T T F T F F T T F T T T T T T F F T F F T F T T T T T T F T F T F F T F F T F T T F F F F F T F T Argument is invalid.
133) ...Sharon is not a competition swimmer. 134) ...some taxi drivers are not on strike. 135) A 136) B 137) A 138) B 139) B 140) A
32