Prestatistics
Review #1
List the elements in the set.
1) The set of the days of the week
A) {Sunday, Monday, Tuesday, Wednesday, Thursday,
Friday, Sunday}
B) {Tuesday, Thursday}
C) {Friday, Monday, Saturday, Sunday, Thursday,
Tuesday, Wednesday}
D) {Saturday, Sunday}
Express the set using the roster method.
2) {x | x is a city in the country where you live}
The bar graph shows the percentage of adults that use the Internet for specific tasks. Use the graph to represent the
given set using the roster method.
3)
29
24
20
13
8
{x | x is a task in which usage lies between 10% and 28%}
A) {information searches, news, job}
C) {news}
Express the set using the roster method.
4) the set of odd natural numbers less than 19
A) {1, 3, 5, . . . , 17}
B) {0, 1, 3, 5, . . . , 17}
B) {news, job}
D) {email, information searches, news, job, school}
C) {2, 4, 6, . . . , 18}
D) {1, 3, 5, . . . , 19}
5) {x | x ∈ N and x is greater than 1}
B) {2,3,4}
A) {1,2,3,...}
C) {2,3,4,...}
D) {2,4,6,...}
6) {x | x ∈ N and x lies between 2 and 6}
B) {2, 3, 4, 5}
A) {2, 3, 4, 5, 6}
C) {3, 4, 5}
D) {3, 4, 5, 6}
Express the set using set-builder notation. Use inequality notation to express the consition x must meet in order to be a
member of the set.
7) A = {18, 19, 20, 21, 22,...}
A) {x | x ∈ N and x ≥ 18}
B) {x | x ∈ N and x ≤ 18}
C) {x | x ∈ N and x ≥ 22}
D) {x | x ∈ N and x > 18}
Find the cardinal number for the set.
8) {x | x is a day of the week that begins with the letter N}
A) 0
B) 3
C) 1
D) 2
Are the sets equivalent?
9) A is the set of residents age 70 or older living in the United States
B is the set of residents age 70 or older registered to vote in the United States
A) Yes
B) No
Determine whether the set is finite or infinite.
10) {x | x ∈ N and x ≥ 1000}
A) Finite
11) The set of natural numbers less than 1
A) Finite
Are the sets equal?
12) {50, 52, 54, 56, 58} = {52, 54, 56, 58}
A) Yes
13) A = {17, 19, 21, 23, 25}
B = {18, 20, 22, 24, 26}
A) Yes
B) Infinite
B) Infinite
B) No
B) No
Determine whether the statement is true or false.
14) ∅ ⊆ {France, Germany, Switzerland}
A) True
B) False
List all the subsets of the given set.
15) {8}
A) { }
C) {8}, { }
B) {8}
Calculate the number of subsets and the number of proper subsets for the set.
16) the set of natural numbers less than 10
A) 511; 512
B) 511; 510
C) 512; 511
D) {0}, {8}, { }
D) 510; 511
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.
17) Let M = the set of Independent men who like classical music
W = the set of Independent women who like classical music
If x ∈ I, then x ∈ W.
A) True
B) False
18) Let M = the set of Republican men who like classical music
W = the set of Republican women who like classical music
If x ∈ M, then x ∈ R.
A) True
B) False
19) If x ∈ D, then x∉ R.
A) True
B) False
20) Let M = the set of Independent men who like classical music
W = the set of Independent women who like classical music
The set of elements in M and W combined is equivalent to set I.
A) True
B) False
Describe a universal set U that includes all elements in the given sets. Answers may vary.
21) A = {fruit juice, coffee}
B = {tea, spring water}
Use the Venn diagram to list the elements of the set in roster form.
22) List the elements of B.
A) {14, 16, 17}
B) {13, 17}
C) {12, 13, 15, 16, 17}
D) {11, 14}
Use the following definition to place the indicated natural number in the correct region of the Venn diagram.
A palindromic number is a natural number whose value does not change if its digits are reversed.
U = the set of natural numbers
A = the set of palindromic numbers
B = the set of odd numbers
23) 110
A)
B)
110
110
C)
D)
110
110
24) 276
A)
B)
276
276
C)
D)
276
276
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.
25) Cʹ
B) Cʹ = {21, 23, 25, 27, ..., 39}
A) Cʹ = {22, 24, 26, 28, ...}
C) Cʹ = {21, 23, 25, 27, ...}
D) Cʹ = {21, 22, 23, 24, ...}
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 ∪ B)ʹ
A) {t, v, x}
B) {s, u, w}
27) B ∪ C
A) {q, s, u, w, y}
C) {q, r, s, t, u, v, w, x, y, z}
28) C ∪ ∅
A) {q, s, u, w, y}
C) {r, t, v, x}
D) {r, s, t, u, v, w, x, z}
B) {v, w, x, y, z}
D) {q, s, v, w, x, y, z}
B) {v, w, x, y, z}
C) {q, s, y, z}
D) { }
29) Aʹ ∪ B
A) {q, s, t, u, v, w, x, y}
C) {q, r, s, t, v, x, y, z}
B) {s, u, w}
D) {r, s, t, u, v, w, x, z}
30) Cʹ ∪ Aʹ
A) {s, t}
C) {q, r, s, t, u, v, x, z}
B) {w, y}
D) {q, s, u, v, w, x, y, z}
31) C ∪ ∅
A) { }
B) {q, s, u, w, y}
C) {q, s, y, z}
D) {v, w, x, y, z}
C) {11, 14, 18, 19}
D) {12, 15, 16, 18, 19}
Use the Venn diagram to list the elements of the set in roster form.
32)
Aʹ
A) {12, 15, 16}
B) {11, 13, 14, 17}
Use the formula for the cardinal number of the union of two sets to solve the problem.
33) 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.
34) (A ∪ B) ∩ (A ∪ C)
A) {q, s, u, w, y}
B) {r, t, v, x}
35) (A ∩ B) ∪ (A ∩ C)
A) {q, s, u, w, y}
36) (A ∩ B ∩ C)ʹ
A) {q, r, s, t, u, v, w, x, z}
C) {q, s, u, w, z}
B) {q, s, v, w, y}
C) {q, s, w, y}
D) {q, s, u, w, y, z}
C) {r, t, u, v, x, z}
D) {q, s, w, y}
B) {r, t, v, x}
D) ∅
Use the Venn diagram shown to answer the question.
37) Which regions represent set D ∪ F?
A) VIII
C) I, II, IV, V, VI, VII, VIII
B) I, II, IV, V, VI, VII
D) III
Use set notation to identify the shaded region.
38)
A) A ∩ B
B) A - B
C) A ∪ B
D) A ∩ B
A) Bʹ ∩ (A ∪ C)
B) (A ∪ C) ∩ Bʹ
C) (C ∩ Bʹ) ∪ A
D) A ∪ C
39)
The chart shows the most common causes of death in certain areas of the United States.
Most Common Causes of Death in U.S.
Region A
Region B
Region C
1. heart disease
1. heart disease
1. heart disease
2. cerebrovascular 2. cerebrovascular 2. cerebrovascular
3. COPD
3. COPD
3. COPD
4. pneumonia
4. accidents
4. accidents
5. accidents
5. pneumonia
5. liver disease
Use the Venn diagram to indicate in which region each cause should be placed.
40) pneumonia
A) VI
B) V
C) IV
D) II
Use the Venn diagram shown below to solve the problem.
41) a) Which regions are represented by (Aʹ ∩ B)ʹ?
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?
42) a) Which regions are represented by (Aʹ ∪ B)ʹ?
b) Which regions are represented by Aʹ ∩ B?
c) Based on parts a) and b), what can you conclude about the relationship between (Aʹ ∪ B)ʹ and Aʹ ∩ B?
Use the accompanying Venn diagram that shows the number of elements in regions I through IV to answer the question.
43)
7
13
12
8
How many elements belong to set A but not set B?
A) 12
B) 8
C) 13
D) 7
C) 29
D) 9
C) 9
D) 10
44)
9
15
14
10
How many elements belong to set A or set B?
A) 48
B) 38
45)
9
17
16
10
How many elements belong to set A and set B?
A) 42
B) 33
Use the given cardinalities to determine the number of elements in the specific region.
46) 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
47) 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 VIII.
A) 54
B) 48
C) 34
D) 0
Use a Venn diagram to answer the question.
48) At East Zone University (EZU) there are 565 students taking College Algebra or Calculus. 327 are 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) 238
B) 537
C) 271
D) 299
A pollster conducting a telephone poll asked three questions:
1. Are you religious?
2. Have you spent time with a person convicted of a crime?
3. Are you in favor of the death penalty?
Solve the problem.
49) Construct a Venn Diagram with three circles that can assist the pollster in tabulating the responses to the three
questions.
50) Write the letter c in every region of the diagram that represents the people polled who do not consider
themselves religious, who have not spent time with aperson convicted with a crime, and who are in favor of the
death penalty.
A)
B)
C)
D)
Form the negation of the statement.
51) Warsaw is not the capital of Italy.
A) It is not true that Italy is not the capital of Warsaw.
B) It is true that Warsaw is not the capital of Italy.
C) It is not true that Warsaw is not the capital of Italy.
D) It is true that Italy is not the capital of Warsaw.
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.
52) The commute to work is long.
A) r
B) ~r
C) ~s
D) s
Express the symbolic statement ~p in words.
53) p: Lake Mead is one of the Great Lakes.
A) It is true that Lake Mead is one of the Great Lakes.
B) Lake Mead is not one of the Great Lakes.
C) Lake Mead is truly one of the Great Lakes.
D) It is not true that Lake Mead 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.
54) Some violinists are not humans.
A) Some humans are not violinists.
B) Not all violinists are humans.
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)
55) Some drinks are not liquids.
A) All drinks are liquids.
B) No drinks are liquids.
C) All liquids are drinks.
D) All drinks are not liquids.
56) All athletes are famous.
A) All athletes are somewhat famous.
C) Some athletes are famous.
B) Some athletes are not famous.
D) All athletes are not famous.
Choose the correct conclusion.
57) In a statement to the press, a representative of a tobacco company stated ʺIn our years of research, we never
discovered a link between cancer and smoking.ʺ That statement was later found to be false. Therefore, in their
years of research:
A) The tobacco company sometimes discovered a link between cancer and smoking.
B) The tobacco company sometimes did not discover a link between cancer and smoking.
C) The tobacco company always discovered a link between cancer and smoking.
D) The tobacco company never discovered a link between cancer and smoking.
List the simple statements contained in the quotation and assign each one a letter. Then rewrite the compound statement
in symbolic form.
58) ʺIf he does that, I will scream, but if he doesnʹt, I wonʹt.ʺ
Given that p and q each represents a simple statement, write the indicated symbolic statement in words.
59) p: The refrigerator is working.
q: The milk is warm.
~ p ∧ q
A) The refrigerator is not working and the milk is warm.
B) If the milk is warm, then the refrigerator is not working.
C) The refrigerator is not working if and only if the milk is warm.
D) The refrigerator is working and the milk is warm.
60) p: The fan is working.
q: The bedroom is stuffy.
p ∨ ~ q
A) If the fan is working, then the bedroom is not stuffy.
B) The fan is not working or the bedroom is stuffy.
C) If the fan is not working, then the bedroom is 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.
61) I change the station if and only if itʹs not true that both I like the song and the DJ is entertaining.
B) r ↔ ~(p ∧ q)
C) (r ↔ ~p) ∧ q
D) r ↔ (~p ∧ q)
A) r ↔ ~(p ∨ q)
Use grouping symbols to clarify the meaning of the symbolic statement.
62) p ↔ r ∧ q → p ∨ r
A) [p ↔ (r ∧ q)] → (p ∨ r)
B) (p ↔ r) ∧ [q → (p ∨ r)]
D) [(p ↔ r) ∧ q] → (p ∨ r)
C) p ↔ [(r ∧ q) → (p ∨ r)]
Write the statement in symbolic form to determine the truth value for the statement.
63) 5 × 2 = 10 or French is a language.
A) True
B) False
Complete the truth table by filling in the required columns.
64) ~ 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 F F T
T F F T F
F T T F F
F F T T F
C)
B)
p q
T T
T F
F T
F F
~ p
F
F
T
T
~ q
F
F
T
F
~ p ∨ ~ q
F
T
T
T
p q
T T
T F
F T
F F
~ p
F
F
T
T
~ q
F
T
F
T
~ p ∨ ~ q
F
T
T
T
D)
p q
T T
T F
F T
F F
~ p
F
F
T
T
~ q
F
T
F
T
~ p ∨ ~ q
F
F
F
T
Construct a truth table for the statement.
65) (r ∧ q) ∨ (~r ∧ ~q)
A) r
q
(r ∧ q) ∨ (~r ∧ ~q)
B) r
q
(r ∧ q) ∨ (~r ∧ ~q)
T
T
F
F
C) r
T
F
T
F
q
F
F
T
T
(r ∧ q) ∨ (~r ∧ ~q)
T
T
F
F
D) r
T
F
T
F
q
T
T
T
F
(r ∧ q) ∨ (~r ∧ ~q)
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
F
F
66) ~(~(r ∨ p))
A) r
p
~(~(r ∨ p))
B) r
p
~(~(r ∨ p))
T
T
F
F
C) r
T
F
T
F
p
T
T
T
F
~(~(r ∨ p))
T
T
F
F
D) r
T
F
T
F
p
T
T
F
F
~(~(r ∨ p))
T
F
F
T
T
F
T
T
F
F
T
F
T
F
F
F
F
T
67) ~(q ∨ t) ∧ ~(t ∧ q)
A) q
t
~(q ∨ t) ∧ ~(t ∧ q)
B) q
t
~(q ∨ t) ∧ ~(t ∧ q)
T
T
F
F
C) q
T
F
T
F
t
F
F
F
T
~(q ∨ t) ∧ ~(t ∧ q)
T
T
F
F
D) q
T
F
T
F
t
F
F
T
F
~(q ∨ t) ∧ ~(t ∧ q)
T
T
F
F
T
F
T
F
F
T
T
F
T
T
F
F
T
F
T
F
F
F
F
F
68) (p ∧ w) ∧ (~w ∨ t)
A) p
w
t
(p ∧ w) ∧ (~w ∨ t)
B) p
w
t
(p ∧ w) ∧ (~w ∨ t)
T
T
T
T
F
F
F
F
C) p
T
T
F
F
T
T
F
F
w
T
F
T
F
T
F
T
F
t
F
T
T
T
T
F
T
T
(p ∧ w) ∧ (~w ∨ t)
T
T
T
T
F
F
F
F
D) p
T
T
F
F
T
T
F
F
w
T
F
T
F
T
F
T
F
t
F
T
T
F
T
F
F
T
(p ∧ w) ∧ (~w ∨ t)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
T
F
T
T
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
F
F
F
F
F
F
F
Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound
statement.
69) p ∧ q
A) True
B) False
70) [(~p ∧ ~q) ∨ ~q]
A) True
B) False
Use the information in the graphs to determine the truth value of the compound statement.
71) It is not true that in 1995, 43.9% of declared majors were science or 11.5% were undecided.
A) True
B) False
72) From 1985 though 2005, the percentage of science majors increased or the percentage of liberal arts majors
increased, and those who where undecided did not decrease.
A) True
B) False
Construct a truth table for the statement.
73) (p ∧ q) → (p ∨ q)
A)
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
C)
B)
p q
T T
T F
F T
F F
p ∧ q
T
T
T
F
p ∨ q
T
F
F
F
(p ∧ q) → (p ∨ q)
T
F
F
T
p q
T T
T F
F T
F F
p ∧ q
T
F
F
F
p ∨ q
T
T
T
F
(p ∧ q) → (p ∨ q)
T
T
T
T
D)
p q
T T
T F
F T
F F
p ∧ q
T
F
F
T
p ∨ q
T
T
T
F
(p ∧ q) → (p ∨ q)
T
T
T
F
74) (~p ∨ ~q) → ~(q ∧ p)
A) p
q
(~p ∨ ~q) → ~(q ∧ p)
B) p
q
(~p ∨ ~q) → ~(q ∧ p)
T
T
F
F
C) p
T
F
T
F
q
T
F
F
T
(~p ∨ ~q) → ~(q ∧ p)
T
T
F
F
D) p
T
F
T
F
q
T
T
T
F
(~p ∨ ~q) → ~(q ∧ p)
T
T
F
F
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
F
F
F
F
75) ~(p ∧ q) → ~(p ∨ q)
A) p
q
~(p ∧ q) → ~(p ∨ q)
B) p
q
~(p ∧ q) → ~(p ∨ q)
T
T
F
F
C) p
T
F
T
F
q
T
T
T
T
~(p ∧ q) → ~(p ∨ q)
T
T
F
F
D) p
T
F
T
F
q
F
F
F
T
~(p ∧ q) → ~(p ∨ q)
T
T
F
F
T
F
T
F
T
T
F
F
T
F
T
F
T
F
F
T
F
T
T
T
Construct a truth table for the given statement and then determine if the statement is a tautology.
76) [ (p ∧ ~ q) ∧ ~ p ] → ~ q
A)
p q ~ q p ∧ ~ q ~ p
(p ∧ ~ q) ∧ ~ p [ (p ∧ ~ q) ∧ ~ p ] → ~ q
T T F F
F
F
F
Is not a tautology.
T F T T
F
F
F
F T F F
T
F
F
F F T F
T
F
F
B)
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ∧ ~ q
F
T
F
F
~ p
F
F
T
T
(p ∧ ~ q) ∧ ~ p
F
F
F
F
[ (p ∧ ~ q) ∧ ~ p ] → ~ q
T
Is not a tautology.
F
T
F
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ∧ ~ q
F
T
F
F
~ p
F
F
T
T
(p ∧ ~ q) ∧ ~ p
F
F
F
F
[ (p ∧ ~ q) ∧ ~ p ] → ~ q
T
Is a tautology.
T
T
T
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ∧ ~ q
F
T
F
F
~ p
F
F
T
T
(p ∧ ~ q) ∧ ~ p
F
F
F
F
[ (p ∧ ~ q) ∧ ~ p ] → ~ q
T
Is not a tautology.
T
T
T
C)
D)
Construct a truth table for the given statement.
77) ~ (p ↔ ~ q)
A)
p q ~ q p ↔ ~ q ~ (p ↔ ~ q)
T T F F
T
T F T T
F
F T F T
F
F F T T
F
C)
B)
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
F
T
T
F
~ (p ↔ ~ q)
T
F
F
T
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
T
F
F
T
~ (p ↔ ~ q)
F
T
T
F
D)
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
F
T
T
F
78) (p ↔ ~ q) → ( ~q → ~ p)
A)
p q ~ q p ↔ ~ q
T T F F
T F T T
F T F T
F F T F
~ (p ↔ ~ q)
T
T
T
T
~ p
F
F
T
T
~q → ~ p
T
F
T
T
(p ↔ ~ q) → ( ~q → ~ p)
T
T
T
T
B)
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
F
T
T
F
~ p
F
F
T
T
~q → ~ p
T
F
T
T
(p ↔ ~ q) → ( ~q → ~ p)
T
F
T
T
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
F
T
T
F
~ p
F
F
T
T
~q → ~ p
T
F
F
T
(p ↔ ~ q) → ( ~q → ~ p)
T
F
F
T
p q
T T
T F
F T
F F
~ q
F
T
F
T
p ↔ ~ q
F
T
T
F
~ p
F
F
T
T
~q → ~ p
T
F
T
T
(p ↔ ~ q) → ( ~q → ~ p)
F
F
T
F
C)
D)
79) [ (p ∧ q) ∧ (p → q) ] ↔ (p ∨ q )
A)
p q p ∧ q p → q (p ∧ q) ∧ (p → q) T T T T
T
T F F F
F
F T F T
F
F F F T
F
p ∨ q
T
T
T
F
[ (p ∧ q) ∧ (p → q) ] ↔ (p ∨ q )
T
F
F
T
B)
p q
T T
T F
F T
F F
p ∧ q
T
F
F
F
p → q
T
F
T
T
(p ∧ q) ∧ (p → q) T
F
T
T
p ∨ q
T
T
T
F
[ (p ∧ q) ∧ (p → q) ] ↔ (p ∨ q )
T
F
T
T
p q
T T
T F
F T
F F
p ∧ q
T
F
F
F
p → q
T
F
T
T
(p ∧ q) ∧ (p → q) T
F
F
F
p ∨ q
T
F
F
F
[ (p ∧ q) ∧ (p → q) ] ↔ (p ∨ q )
T
T
T
T
p q
T T
T T
F T
F F
p ∧ q
T
F
F
F
p → q
T
F
T
T
(p ∧ q) ∧ (p → q) T
F
T
F
p ∨ q
T
F
T
F
[ (p ∧ q) ∧ (p → q) ] ↔ (p ∨ q )
T
F
F
F
C)
D)
Construct a truth table for the statement. Then determine if the statement is a tautology.
80) (q → p) ↔ (~ p ∨ q)
A)
p q q → p ~ p ~ p ∨ q (q → p) ↔ (~ p ∨ q)
T T T
T T
F
Is not a tautology.
F
T F T
F F
F T F
T T
F
F F F
T T
T
B)
p q
T T
T F
F T
F F
q → p
T
T
F
T
~ p
F
F
T
T
~ p ∨ q
T
F
T
T
(q → p) ↔ (~ p ∨ q)
T
Is not a tautology.
F
F
T
p q
T T
T F
F T
F F
q → p
T
T
F
T
~ p
F
F
T
T
~ p ∨ q
T
F
T
T
(q → p) ↔ (~ p ∨ q)
T
Is not a tautology.
F
T
T
p q
T T
T F
F T
F F
q → p
T
F
T
T
~ p
F
F
T
T
~ p ∨ q
T
F
T
T
(q → p) ↔ (~ p ∨ q)
T
Is a tautology.
T
T
T
C)
D)
Use a truth table to determine whether the two statements are equivalent.
81) ~p ∧ ~q and ~(p ∨ q)
A) Yes
B) No
82) q → p and ~q ∨ p
A) Yes
B) No
83) ~(q → p) and q ∧ ~p
A) Yes
Select the statement that is equivalent to the given statement.
84) I ate a pickle or a hot dog.
A) If I did not eat a pickle, then I ate a hot dog.
B) If I ate a pickle, then I ate a hot dog.
C) If I did not a pickle, then I did not eat a hot dog.
D) If I ate a pickle, then I did not eat a hot dog.
B) No
Write the negation of the conditional statement.
85) If there is an earthquake, then all nurses are on call.
A) There is an earthquake and some nurses are not on call.
B) If there is an earthquake, then some nurses are not on call.
C) There is an earthquake and no nurses are on call.
D) There is not an earthquake and some nurses are not on call.
86) If I get a high-paying job, then I can pay off all my bills.
A) I donʹt get a high-paying job and cannot pay off all my bills.
B) I donʹt get a high-paying job and can pay off all my bills.
C) I get a high-paying job and I cannot pay off all my bills.
D) I get a high-paying job and can pay off all my bills.
Use a truth table to determine whether the symbolic form of the argument is valid or invalid.
87) p → q
~ p
∴ ~ q
A)
p q
T T
T F
F T
F F
p→q ~ p (p → q) ∧ ~ p ~ q
T F F
F
F F F
T
F T F
F
F T F
T
[ (p → q) ∧ ~ p ] → ~ q
T
The argument is valid.
T
T
T
p q
T T
T F
F T
F F
p→q ~ p (p → q) ∧ ~ p ~ q
T F F
F
F F T
T
T T T
F
T T T
T
[ (p → q) ∧ ~ p ] → ~ q
T
The argument is invalid.
T
F
T
p q
T T
T F
F T
F F
p→q ~ p (p → q) ∧ ~ p ~ q
T F T
F
F F F
T
F T T
F
F T T
T
[ (p → q) ∧ ~ p ] → ~ q
T
The argument is valid.
T
T
T
p q
T T
T F
F T
F F
p→q ~ p (p → q) ∧ ~ p ~ q
T F F
F
F F F
T
T T T
F
T T T
T
[ (p → q) ∧ ~ p ] → ~ q
T
The argument is invalid.
T
F
T
B)
C)
D)
88) q → p
p → r
∴ r → q
A)
p q r q → p p → r (q → p) ∧ (p → r) r → q
T T T T T T
T
T T F T F F
T
F
T F T T T T
T F F T F F
T
F T T F T F
T
F T F F T F
T
F F T T T T
F
F F F T T T
T
Symbolic argument is invalid.
[(q → p) ∧ (p → r)] ∧ (r → q)
T
F
F
F
F
F
F
T
p q r q → p p → r (q → p) ∧ (p → r) r → q
T T T T T T
T
T T F T F F
T
T F T T T F
T
T F F T F F
T
F T T F T F
T
F T F F T F
T
F F T T T F
T
F F F T T F
T
Symbolic argument is valid.
[(q → p) ∧ (p → r)] → (r → q)
T
T
T
T
T
T
T
T
p q r q → p p → r (q → p) ∧ (p → r) r → q
T T T T T T
T
T T F T F F
T
T F T T T F
T
T F F T F F
T
F T T F T F
T
F T F F T F
T
F F T T T F
T
F F F T T T
T
Symbolic argument is valid.
[(q → p) ∧ (p → r)] → (r → q)
T
T
T
T
T
T
T
T
p q r q → p p → r (q → p) ∧ (p → r) r → q
T T T T T T
T
T T F T F F
T
T F T T T T
F
T F F T F F
T
F T T F T F
T
F T F F T F
T
F F T T T T
F
F F F T T T
T
Symbolic argument is invalid.
[(q → p) ∧ (p → r)] → (r → q)
T
T
F
T
T
T
F
T
B)
C)
D)
89) p → q
q ∴ p
A) Valid
B) Invalid
90) p → q
~q ∴ ~p
A) Valid
B) Invalid
Draw a valid conclusion from the given premises.
91) All birds have wings. None of my pets are birds. All animals with wings can flap them.
Therefore....
A) All my pets can flap their wings.
B) All birds can flap their wings.
C) No birds can flap their wings.
D) None of my pets can flap their wings.
Use an Euler diagram to determine whether the argument is valid or invalid.
92) All birds have feathers.
No mammal has feathers.
Therefore, no mammals are birds.
A) valid
B) invalid
93) All rock stars are performers.
Mauricio is a rock star.
Therefore, Mauricio is a performer.
A) valid
B) invalid
94) All horses whinny.
Some horses are brown.
Therefore, all brown horses whinny.
A) valid
B) invalid
95) Not all that glitters is gold.
My ring glitters. Therefore, My ring is not gold.
A) valid
B) invalid
96) All students who study get better grades.
Roger is a student who studies. Therefore, Roger will get better grades.
A) valid
B) invalid
97) No even number is divisible by 3.
18 is an even number. Therefore, 18 is not divisible by 3.
A) valid
B) invalid
98) All tigers are felines.
All felines are mammals.
All mammals nurse their young.
Therefore, All tigers nurse their young.
A) valid
B) invalid
99) 10 is a real number.
100 is a real number.
Therefore, 10 is 100.
A) valid
B) invalid
100) Rational numbers are real numbers.
Integers are rational numbers. Therefore, integers are real numbers.
A) valid
Find the least common multiple of the numbers.
101) 245 and 420
A) 2940
B) 8575
102) 1250 and 1800
A) 62,500
B) 45,000
B) invalid
C) 102,900
D) 14,700
C) 90,000
D) 2,250,000
Solve.
103) A perfect number is a natural number that is equal to the sum of its factors, excluding the number itself.
Determine whether or not the number 8 is perfect.
A) Yes
B) No
Insert < or > in the area between the integers to make the statement true.
104) -11 -8
A) -11 > -8
B) -11 < -8
Find the quotient, or, if applicable, state that the expression is undefined.
105) (-54) ÷ (9)
A) 9
B) 6
C) -6
D) The expression is undefined.
Use the order of operations to find the value of the expression.
106) 6 - 3(9 - 7) - 7
A) 13
B) 5
C) -1
D) -7
107) (2 + 1)3 - (3 - 1)3
A) 19
B) -19
C) -37
D) 37
108) 7(5 - 3)3 - 2(6 - 4)3
A) -20
B) -40
C) 40
D) 20
109) 4 2 - 8 ÷ 2 2 · 4 - 4
A) 4
B) -2
C) 0
D) 12
Reduce the rational number to its lowest terms.
52
110)
117
A)
52
117
B)
13
9
C)
4
13
D)
4
9
C)
409
19
D)
409
10
Convert the mixed number to an improper fraction.
10
111) 21
19
A)
589
10
B)
210
19
Convert the improper fraction to a mixed number.
50
112)
3
A) 17
2
3
B)
2
3
C) 15
2
7
D) 16
2
3
Express the rational number as a decimal.
4
113)
5
A) 0.1
B) 1.6
C) 1
D) 0.8
Express the rational number as a decimal. Then insert either < or > between the rational number to make the statement
true.
11
3
114) - - 19
17
A) <
B) >
Express the terminating decimal as a quotient of integers. If possible, reduce to lowest terms.
115) 0.6
2
3
2
B)
C)
A)
3
50
33
D)
3
5
Perform the indicated operation(s). Where possible, reduce the answer to lowest terms.
1
116)
- 8
3
A)
117) - 8
3
B) - 1
24
C) - 8
3
D) - 7
3
B) - 4
31
C)
3
31
D) - 2
105
3
1
- 10
21
A)
1
70
118)
9
2
- - 21
21
A)
1
3
B)
1
2
C)
11
21
D)
11
7
7
8
D)
3
4
1
24
D)
5
3
Perform the indicated operations. If possible, reduce the answer to its lowest terms.
1 1 3
119) + · 2 8 2
A)
120)
5
16
B)
15
16
C) 1
B)
1
8
C)
2 1
4 1
- ÷ - 3 6
5 2
A)
1
9
Solve the problem.
3
121) A recipe calls for 1 cups of sugar for 18 brownies. How much of sugar is needed to make 12 brownies?
5
2
A) 2 c
5
1
B) 19 c
5
C)
4
c
45
Perform the indicated operation. Simplify the answer when possible.
122) 4 5 + 14 5
A) 18 5
B) -19 5
C) -10 5
123)
3 + 12
A) 3 3
B) 12
D) 1
1
c
15
D) 11 5
C) 6
D) 5 3
Complete the statement to illustrate the commutative property.
124) 5 + (4 + 6) = 5 + (6 + ___ )
A) 4
B) 6
C) 5
D) 10
Complete the statement to illustrate the associative property.
125) (8 + 6) + 4 = ___ + (6 + ___ )
A) 3 and 9
B) 4 and 8
C) 6 and 7
D) 8 and 4
Use the distributive property to simplify the radical expressions.
126) 2( 10 + 2)
A) 3 5
B) 2 5
C) 2 5 + 2
D) 12
State the name of the property illustrated.
127) 6(-5 + 7) = -30 + 42
A) distributive property of multiplication over addition
B) commutative property of addition
C) commutative property of multiplication
D) associative property of addition
Determine if the statement is true or false. Do not use a calculator.
128) 18(180 + 36) = 180 + 36(18)
A) True
129) 160 · 18 + 34 · 18 = (160 + 34) · 18
A) True
B) False
B) False
Use properties of exponents to simplify the expression. First, express the answer in exponential form. Then, evaluate the
expression.
130) (1 7 )8
A) 1 56; 1
B) 1 56; 56
C) 1 15; 1
D) 1 56; 0
Use properties of exponents to simplify the expression. Express answer in exponential form.
131) 4 -3 · 4
A) 4 -2
B) 4 -3
C) -4 2
D) -3 · 4 2
Use properties of exponents to simplify the expression. Express answers in exponential form with positive exponents
only. Assume that any variables in denominators are not equal to zero.
132) (-5x2 y-4 )(-4x2 y2 )
20x4
y2
20x
y6
C) 20xy4
D) 20x6 y4
B) 0.0007329
C) 0.000007329
D) 0.00007329
134) 8.471 × 10-6
A) -8,471,000
B) 0.00008471
C) 0.0000008471
D) 0.000008471
135) 4 × 10 5
A) 4000
B) 0.00004
C) 400,000
D) 200
136) 1 × 10 3
A) 30
B) 10,000
C) 1000
D) 100
C) 2.05011 × 10-7
D) 2.05011 × 106
B) 38 × 10 2
C) 3.8 × 10 2
D) 38 × 10
B) 5.1 × 10 8
C) 5.1 × 10 9
D) 51 × 10 7
B) 37 × 10 -1
C) 3.7 × 10 -3
D) 37 × 10 -4
A)
B)
Express the number in decimal notation.
133) 7.329 × 10-5
A) -732,900
Express the number in scientific notation.
137) 0.000000205011
A) 2.05011 × 10-6
B) 2.05011 × 107
138) 380
A) 3.8 × 10 3
139) 510,000,000
A) 5.1 × 10 7
140) 0.0037
A) 3.7 × 10 -4
Evaluate the algebraic expression for the given value(s) of the variable(s).
141) 6xy ; x = -5, y = -1
A) -30y
B) -30
C) 29
142)
x - 8
; x = -3
x - 2
A)
143)
5
11
B) - 5
C)
11
5
D) - B)
7
12
C)
1
3
D)
B)
19
23
C) - 1
5
x + 4
; x = 3
4x + 8
A)
144)
D) 30
7
20
20
7
x2 - 10x + 5
; x = 4
x2 + 2x - 1
A)
145) 7y + 61
23
19
25
D) - 19
23
55
; x = 5, y = 7
x
A) 60
B) 104
C) 11
D) 18
Simplify the algebraic expression.
146) 20x2 - 20x2
A) -40x2
C) x2
147) 10(9x + 6)
A) 19x + 16
B) 0
D) cannot be simplified
B) 150x
C) 90x + 6
D) 90x + 60
148) [-8x2 + (-3x2 + 4)] + [-(4x2 + (-9 - 5x2 )) - 10x2 ]
B) -22x2 - 5
A) -2x2 - 5
C) -24x2 - 13
D) -20x2 + 13
149) 9 times the product of a number and negative 3
A) 9(x) + 3; -9x + 3
B) 9[x(-3)]; -27x
C) 9[x(3)]; 27x
D) 9(x) + (-3); 9x - 3
150) 19 decreased by 4 times the sum of 2 and a number
A) 19 - [-4(2 + x)]; 27 + 4x
C) 19 - [4(2 + x)]; 4x - 11
B) 19 - [4(2 + x)]; 11 - 4x
D) 19 - [-4(2 + x)]; 27 - 4x
Express the fraction as a percent.
5
151)
8
A) 80%
B) 6.3%
C) 78.1%
D) 62.5%
152)
1
4
A) 80%
B) 2.5%
C) 25%
D) 31.3%
B) 97.5 %
C) 10.26 %
D) 1.03 %
B) 1.63 %
C) 16.25 %
D) 61.54 %
B) 8.3%
C) 0.083%
D) 83%
156) 3.8
A) 0.38%
B) 0.0038%
C) 380%
D) 38%
157) 0.00952
A) 0.0952%
B) 0.476%
C) 0.000952%
D) 0.952%
B) 0.42%
C) 4200%
D) 4.2%
B) 0.0764
C) 0.654
D) 7.64
160) 580%
A) 5.81
B) 0.58
C) 58
D) 5.8
161) 6%
A) 0.6
B) 0.06
C) 60
D) 6
B) 7.1429
C) 0.00714
D) 0.71429
153)
39
40
A) 9.75 %
154)
13
80
A) 6.15 %
Write the decimal as a percent.
155) 0.083
A) 0.0083%
158) 42
A) 2100%
Express the percent as a decimal.
159) 76.4%
A) 0.764
162)
5
%
7
A) 0.00071
Answer Key
Testname: PRESTAT REVIEW 1 NEW
1) C
2) Answers may vary. Possible answers are:
{Chicago}, {Los Angeles}, {New York City}, and {Springfield}
3) A
4) A
5) C
6) C
7) A
8) A
9) B
10) B
11) A
12) B
13) B
14) A
15) C
16) C
17) B
18) A
19) A
20) A
21) Answers may vary. One possible answer is: U = the set of all non-carbonated beverages.
22) C
23) B
24) A
25) C
26) C
27) D
28) B
29) C
30) C
31) D
32) D
33) B
34) D
35) D
36) A
37) B
38) A
39) C
40) D
c) They are not equal.
41) a) I, II, and IV b) I
b) III c) They are not equal.
42) a) I
43) C
44) B
45) C
46) A
47) C
48) A
Answer Key
Testname: PRESTAT REVIEW 1 NEW
49)
50) D
51) C
52) B
53) B
54) B
55) A
56) B
57) A
58) p: He does that.
q: I scream
(p → q) ∧ (~ p → ~ q)
59) A
60) D
61) B
62) C
63) A
64) D
65) C
66) A
67) A
68) D
69) B
70) A
71) B
72) A
73) D
74) C
75) C
76) C
77) B
78) B
79) A
80) B
81) A
82) A
83) A
84) A
85) A
86) C
87) D
88) D
Answer Key
Testname: PRESTAT REVIEW 1 NEW
89) B
90) A
91) B
92) A
93) A
94) A
95) B
96) A
97) A
98) A
99) B
100) A
101) A
102) B
103) B
104) B
105) C
106) D
107) A
108) C
109) A
110) D
111) C
112) D
113) D
114) A
115) D
116) C
117) A
118) C
119) B
120) A
121) D
122) A
123) A
124) A
125) D
126) C
127) A
128) B
129) A
130) A
131) A
132) A
133) D
134) D
135) C
136) C
137) C
138) C
139) B
Answer Key
Testname: PRESTAT REVIEW 1 NEW
140)
141)
142)
143)
144)
145)
146)
147)
148)
149)
150)
151)
152)
153)
154)
155)
156)
157)
158)
159)
160)
161)
162)
C
D
C
A
D
A
B
D
D
B
B
D
C
B
C
B
C
D
C
A
D
B
C
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