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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.
- 4 ∈ {1, 2, 3, ..., 15} A) True B) False
A) True B) False
A) True B) False
Fill in the blank with either ∈ or ∉ to make the statement true.
- Saskatchewan _____ the set of states in the United States A) ∈ B) ∉
Find the cardinal number for the set.
- {27, 29, 31, 33, 35} A) 5 B) 27 C) 4 D) 6
- Determine the cardinal number of the set {x | x is a letter of the alphabet} A) 23 B) 26 C) 30 D) 25
Are the sets equivalent?
- A = {23, 25, 27, 29, 31} B = {24, 26, 28, 30, 32} A) Yes B) No
8) A = {13, 14, 14, 15, 15, 15, 16, 16, 16, 16}
B = {16, 15, 14, 13}
A) Yes B) No
Are the sets equal?
- {50, 52, 54, 56, 58} = {52, 54, 56, 58} A) Yes B) No
- A is the set of residents age 26 or older living in the United States 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}
B = {21, 20, 19, 18}
A) Yes B) No
12) A = {14, 15, 16, 17, 18}
B = {13, 14, 15, 16, 17}
A) Yes B) No
Write ⊆ or ⊈ in the blank so that the resulting statement is true.
- {4, 34, 39} {15, 34, 39, 49} A) ⊆ B) ⊈
- {c, a, n, d, i, d, a, t, e} _____ {a, c, d, e, i, t, a, n, d} A) ⊆ B) ⊈
Determine whether the statement is true or false.
- ∅ ⊆ {France, Germany, Switzerland} A) True B) False
Use ⊆, ⊈, ⊂, or both ⊂ and ⊆ to make a true statement.
- {a, b} {z, a, y, b, x, c} A) ⊂ and ⊆ B) ⊈ C) ⊂ D) ⊆
List all the subsets of the given set.
- {8} A) {0}, {8}, { } B) { } C) {8} D) {8}, { }
Calculate the number of subsets and the number of proper subsets for the set.
- 1 6
A) 15; 14 B) 14; 15 C) 16; 15 D) 15; 16
A) 64; 63 B) 63; 62 C) 63; 64 D) 62; 63
- {x | x is a day of the week} A) 127; 126 B) 64; 65 C) 128; 127 D) 128; 129
- List the elements of U.
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.
- A = {2, 4, b, d} 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.
- Aʹ 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.
- Cʹ 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.
- 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 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.
- A ∩ Bʹ 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}
29) (A ∩ B)ʹ
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
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
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
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
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
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
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
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
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 ∪ ∅
A) { } B) {q, s, y, z} C) {v, w, x, y, z} D) {q, s, u, w, y}
39) B ∪ U
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}
(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.
- Results of a survey of fifty students indicate that 30 like red jelly beans, 29 like green jelly beans, 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
- Monticello residents were surveyed concerning their preferences for candidates Moore and Allen 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.
- Set A contains 35 elements and set B contains 22 elements. If there are 40 elements in (A ∪ B) then how many elements are in (A ∩ B)? A) 13 B) 17 C) 5 D) 8
- Set A contains 5 elements, set B contains 11 elements, and 3 elements are common to sets A and B. How many elements are in A ∪ B? A) 12 B) 14 C) 16 D) 13
- Set A contains 35 elements and set B contains 22 elements. If there are 40 elements in (A ∪ B) then 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.
- A ∪ (B ∩ C) 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)
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
A) {q, s} B) {v, z} C) {r, t} D) {s, t}
52) (B ∪ C)ʹ ∩ A
A) {w} B) {v} C) ∅ D) {u}
Use the following information to construct a Venn Diagram that illustrates the given sets.
- U = the set of members of the bookclub shown in the chart 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 the bookclub
Numbers of books read
Numbers of books suggested
Year of 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)
- a) Which regions are represented by (A ∪ Bʹ)ʹ? 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.
- a) Which regions are represented by (A ∩ B) ∪ C? 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)?
- a) Which regions are represented by B ∪ (A ∩ C)? 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.
How many elements belong to set B? A) 12 B) 19 C) 25 D) 18
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.
- 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) = 10 Find III.
A) 41 B) 14 C) 30 D) 26
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.
- One does not play hard. A) ~q B) q C) p D) ~p
- The commute to work is long. A) s B) ~r C) r D) ~s
Form the negation of the statement.
- Today is May 10 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.
- p: The Pilgrims did not land in Tahiti. 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.
- p: Vitamin C helps the immune system. 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.
- p: The refrigerator is not working. 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.
- All uncles are males. 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.
- Some flowers are roses. 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.
- Some violinists are not humans. A) Not all violinists are humans. B) Some humans are not violinists. C) All violinists are not humans. D) All violinists are humans.
Write the negation of the quantified statement. (The negation should begin with ʺall,ʺ ʺsome,ʺ or ʺno.ʺ)
- Some mammals are horses. A) All horses are mammals. B) No horses are mammals. C) No mammals are horses. D) Not all mammals are horses.
- No South American soccer teams have won a World Cup. 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.
- All athletes are famous. 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.
- As a special promotion, the Green Thumb organic foods chain said that everyone who came to one 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.
- p: Not all people like football. A) All people like football. B) All people do not like football. C) Some people like football. D) Some people do not like football.
- p: Some athletes are musicians. A) Some athletes are not musicians. B) No athlete is a musician. C) Not all athletes are musicians. D) All athletes are musicians.
- p: Some people donʹt like walking. 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.
- p: Spartacus is a film. q: Rambo is a film. Spartacus is a film and Rambo is a film. A) p ∧ ~ q B) p → q C) p ∨ q D) p ∧ q
- p: They set the alarm. q: They get up on time. They set the alarm or they get up on time. A) p ∨ q B) p ∨ ~ q C) p → q D) p ∧ q
- p: The car has been repaired. 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.
- p: The fan is working. 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.
- If I like the song or the DJ is entertaining then I do not change the station. A) (p ∨ q) → r B) p ∨ (q → r) C) p ∨ (q → ~r) D) (p ∨ q) → ~r
- 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 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.
- Miami is a city and China is a country. A) True B) False
- 5 × 2 = 10 or French is a language. A) True B) False
Complete the truth table by filling in the required columns.
- p ∧ ~ q
p q ~ q p ∧ ~ q T T T F F T F F A) p q ~ q p ∧ ~ q T T F F T F T T F T F F F F T T
B)
p q ~ q p ∧ ~ q T T F T T F T T F T F F F F T F
C) p q ~ q p ∧ ~ q T T F F T F T T F T T F F F T F
D)
p q ~ q p ∧ ~ q T T F F T F T T F T F F F F T F
- ~ (p ∧ q)
p q p ∧ q ~ (p ∧ q) T T T F F T F F A) p q p ∧ q ~ (p ∧ q) T T T F T F F T F T F T F F F F
B)
p q p ∧ q ~ (p ∧ q) T T T T T F F F F T F F F F F F
C) p q p ∧ q ~ (p ∧ q) T T T F T F F T F T F T F F T F
D)
p q p ∧ q ~ (p ∧ q) T T T F T F F T F T F T F F F T
Let p represent a true statement, while q and r represent false statements. Find the truth value of the compound statement.
- ~p ∨ (q ∧ ~r) A) True B) False
- (p ∧ ~q) ∧ r A) True B) False
- ~(~p ∧ ~q) (^) ∨ (~r (^) ∨ ~p) A) True B) False
Construct a truth table for the statement.
- ~ p → ~ q A) p q ~ p ~ q ~ p → ~ q T T F F F T F F T T F T T F F F F T T T
B)
p q ~ p ~ q ~ p → ~ q T T F F T T F F T T F T T F F F F T T T
C) p q ~ p ~ q ~ p → ~ q T T F F T T F F T F F T T F F F F T T T
D)
p q ~ p ~ q ~ p → ~ q T T F T T T F F F T F T T T T F F T F F
- ~(q → ~ p) A) p q ~ p q → ~ p ~(q → ~ p) T T F F T T F F F T F T T T F F F T T F
B)
p q ~ p q → ~ p ~(q → ~ p) T T F F T T F F T F F T T T T F F T T F
C) p q ~ p q → ~ p ~(q → ~ p) T T F T F T F F F T F T T F T F F T F T
D)
p q ~ p q → ~ p ~(q → ~ p) T T F F T T F F T F F T T T F F F T T F
Construct a truth table for the given statement and then determine if the statement is a tautology.
- [ (p → ~ q) ∧ q ] → ~ p 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 F T F T F T T F F T T F T T
Is a tautology.
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 F T F T T T T F F T T F T T
Is a tautology.
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 F T F T T T T F F T T F T T
Is not a tautology.
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 F T F F F T T F F T F F T T
Is a tautology.
