Deck 7: Logic, sets, and Counting

Question 1

Construct a truth table to decide if the two statements are equivalent.
q →p ; p →q

Answer 1

False

Question 2

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false.
{9} ⊂ A

Answer 2

False

Question 3

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement
is true or false.
A ⊂ A

Answer 3

False

Question 4

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement
is true or false.
C ⊆ D

Question 5

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement
is true or false.
{5} ⊆ D

Question 6

Construct a truth table to decide if the two statements are equivalent.
~p K~q; ~(p Vq)

Question 7

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement
is true or false.
U ⊆ A

Question 8

Tell whether the statement is true or false.
0 ∉ Ø

Question 9

Express the proposition as an English sentence and determine whether it is true or false, where p and q are the propositions
p: "9 · 9 = 81" q: "8 · 10 < 7 · 11
The contrapositive of p →q

A)If 8 · 10 is not less than 7 · 11,then 9 · 9 is equal to 81; false
B)If 8 · 10 is less than 7 · 11,then 9 · 9 is equal to 81; false
C)If 8 · 10 is not less than 7 · 11,then 9 · 9 is not equal to 81; false
D)If 8 · 10 is less than 7 · 11,then 9 · 9 is not equal to 81; false

Question 10

Tell whether the statement is true or false.
7 ∉ {14,21,28,35,42}

Question 11

Construct a truth table to decide if the two statements are equivalent.
~(~q); q

Question 12

Tell whether the statement is true or false.
3 ∈ {6,9,12,15,18}

Question 13

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false.
{3,0,8} ⊂ A

Question 14

Provide an appropriate response.
One of the following is false; indicate by letter which one:

A){4} ⊂ {3,4,5,6}
B)4 ∈{3,4,5,6}
C)4 ⊂ {3,4,5,6}
D)5 ∈ {3,4,5,6}

Question 15

Construct a truth table to decide if the two statements are equivalent.
~q ^ p ; ~q →p

Question 16

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false.
{3,0,8} ∈ A

Question 17

Express the proposition as an English sentence and determine whether it is true or false, where p and q are the propositions
p: "9 · 9 = 81" q: "8 · 10 < 7 · 11
p ^q

A)9 · 9 = 81 and 8 · 10 < 7 · 11; false
B)If 9 · 9 = 81,then 8 · 10 < 7 · 11; false
C)8 · 10 is not less than 7 · 11; true
D)9 · 9 = 81 or 8 · 10 < 7 · 11; true

Question 18

Tell whether the statement is true or false.
{all odd integers greater than -3 and less than 5} = {-1,1,3}

Question 19

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false.
{9} ∈ A

Question 20

Construct a truth table to decide if the two statements are equivalent.
~p V~q; ~(p ^ q)

Question 21

Use the Venn diagram to find the requested set.
Find A ⋃ B.

A){d,c,g,i,m,w}
B){d,c,g,i,q,m,w}
C){q}
D){c,g}

Question 22

Use a Venn Diagram and the given information to determine the number of elements in the indicated region.
At Southern States University (SSU)there are 399 students taking Finite Mathematics or Statistics.238 are taking Finite Mathematics,184 are taking Statistics,and 23 are taking both Finite Mathematics and Statistics.How many are taking Finite Mathematics but not Statistics?

A)192
B)215
C)376
D)161

Question 23

Determine whether the given set is disjoint or not disjoint. Consider the set N of positive integers to be the universal set, and let
A = {n ∈N| n > 50}
B = {n ∈N| n < 250}
O = {n ∈N| n is odd}
E = {n ∈N| n is even}
O ⋂ E'

A)not disjoint
B)disjoint

Question 24

Use the Venn diagram to find the requested set.
Find A ⋂ B.

A){a,b,g,j,q,n,w}
B){a,b,g,j,n,w}
C){b,g}
D){q}

Question 25

Use a Venn Diagram and the given information to determine the number of elements in the indicated region.
At Southern States University SSU)there are 719 students taking Finite Mathematics or Statistics.328 are taking Finite Mathematics,476 are taking Statistics,and 85 are taking both Finite Mathematics and Statistics.How many are taking Statistics but not Finite Mathematics?

A)243
B)391
C)158
D)634

Question 26

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let
A = {n ∈N| n > 50}
B = {n ∈N| n < 250}
O = {n ∈N| n is odd}
E = {n ∈N| n is even}
A'

A)finite
B)infinite

Question 27

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let
A = {n ∈N| n > 50}
B = {n ∈N| n < 250}
O = {n ∈N| n is odd}
E = {n ∈N| n is even}
A ⋂ O'

A)finite
B)infinite

Question 28

Let U = {a,l,i,t,e},A = {l,i,t},B = {l,e},C = {a,l,i,t,e},and D = {a,e}.Find (C ⋂ B' ) ⋃ A' .

Question 29

In a marketing survey involving 1,000 randomly chosen people,it is found that 660 use brand P,440 use brand
Q,and 220 use both brands.How many people in the survey use brand P and not brand Q?

Question 30

Use the Venn diagram below to find the number of elements in the region.
n(A ⋃ B)

A)29
B)21
C)11
D)14

Question 31

Use the Venn diagram to find the requested set.
Find A' ⋂ B'.

A){8,3,4,z,r,f}
B){4}
C){8}
D)Ø

Question 32

Use the Venn diagram below to find the number of elements in the region.
n(A ⋂ C)

A)2
B)10
C)18
D)37

Question 33

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let
A = {n ∈N| n > 50}
B = {n ∈N| n < 250}
O = {n ∈N| n is odd}
E = {n ∈N| n is even}
B ⋃ E

A)infinite
B)finite

Question 34

Provide an appropriate response.
Which of the following is NOT a subset of the set {p,o,7}?

A)7
B){o,7}
C)Ø
D){p,o,7}

Question 35

Use the Venn diagram to find the requested set.
Find A.

A){5}
B){7,1,5}
C){7,1,5,x}
D){5,x,q,g}

Question 36

Use the Venn diagram below to find the number of elements in the region.
n(A ⋂ B ⋂ C)

A)44
B)8
C)16
D)18

Question 37

Use a Venn Diagram and the given information to determine the number of elements in the indicated region.
n(A) = 33, n(B) = 19, n(A ⋂ B) = 1, n(A' ⋂ B') = 9. Find n(U).

A)60
B)64
C)51
D)52

Question 38

Use the Venn diagram below to find the number of elements in the region.
n(A)

A)9
B)12
C)4
D)17

Question 39

Determine whether the given set is disjoint or not disjoint. Consider the set N of positive integers to be the universal set, and let
A = {n ∈N| n > 50}
B = {n ∈N| n < 250}
O = {n ∈N| n is odd}
E = {n ∈N| n is even}
A' ⋂ B'

A)not disjoint
B)disjoint

Question 40

Use the Venn diagram below to find the number of elements in the region.
n(C' )

A)29
B)14
C)24
D)39

Question 41

Provide an appropriate response.
License plates are made using 3 letters followed by 3 digits.How many plates can be made if repetition of letters and digits is allowed?

A)308,915,776
B)1,757,600
C)1,000,000
D)175,760
E)17,576,000

Question 42

Provide an appropriate response.
A Super Duper Jean company has 3 designs that can be made with short or long length.There are 5 color patterns available.How many different types of jeans are available from this company?

A)30
B)10
C)8
D)15
E)25

Question 43

Provide an appropriate response.
A local television station sends out questionnaires to determine if viewers would rather see a documentary,an interview show,or reruns of a game show.There were 500 responses with the following results: 150 were interested in an interview show and a documentary,but not reruns.
20 were interested in an interview show and reruns but not a documentary
70 were interested in reruns but not an interview show.
120 were interested in an interview show but not a documentary.
50 were interested in a documentary and reruns.
30 were interested in an interview show and reruns.
40 were interested in none of the three.
How many are interested in exactly one kind of show?

A)230
B)220
C)250
D)240

Question 44

A coin that can turn up either heads (H)or tails (T)is flipped.If a head turns up on the first toss,a spinner that can land on any of the first 7 natural numbers is spun.If a tail turns up,the coin is flipped a second time.What are the different possible outcomes?

Question 45

Provide an appropriate response.
In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take biology,but not history?

A)22
B)5
C)17
D)9

Question 46

Evaluate.
C7,0

A)7
B)6
C)1
D)720

Question 47

Provide an appropriate response.
A survey of residents in a certain town indicates 170 own a dehumidifier,130 own a snow blower,and 80 own a dehumidifier and a snow blower.How many own a dehumidifier or a snow blower?

A)80
B)250
C)170
D)220

Question 48

Provide an appropriate response.
The access code to a house's security system consists of five digits.How many different codes are available if each digit can be repeated?

A)32
B)45
C)5
D)100,000
E)3125

Question 49

Evaluate.
C8,2

A)720
B)28
C)1440
D)4

Question 50

Provide an appropriate response.
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 neither red nor green jelly beans?

A)13
B)12
C)8
D)17

Question 51

Provide an appropriate response.
How many different sequences of 4 digits are possible if the first digit must be 3,4,or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

A)5000
B)2999
C)1512
D)2997
E)2000

Question 52

Provide an appropriate response.
In Virginia,each automobile license plate consists of a single digit followed by three letters,followed by three digits.How many distinct license plates can be formed if there are no restrictions on the digits or letters?

A)17,575,600
B)175,7560,000
C)17,576
D)175,760
E)10,757,600

Question 53

Provide an appropriate response.
Suppose there are 4 trains connecting town X to town Y and 6 roads connecting town Y to town Z.In how many ways can a person travel from X to Z via Y?

A)10
B)16
C)12
D)24
E)36

Question 54

Provide an appropriate response.
A restaurant offered pizza with 3 types of crusts and 7 different toppings.How many different types of pizzas could be offered?

A)21
B)9
C)49
D)10
E)63

Question 55

How many different five-letter code words are possible from the first ten letters of the alphabet if the first letter cannot be a vowel and adjacent letters must be different.

Question 56

A person purchasing a new car has several options: 6 interior color choices,5 exterior color choices,2 choices of radios,and 5 choices of body styles.How many different cars are possible if one choice is made for each option?

Question 57

Determine whether the selection is a permutation, a combination, or neither.
Mary baked 3 pies: 1 for her father,1 for her friend Joe,and 1 for her coworkers.

A)neither
B)combination
C)permutation

Question 58

Provide an appropriate response.
In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take neither biology nor history?

A)8
B)5
C)11
D)22

Question 59

Provide an appropriate response.
Mrs.Bollo's second grade class of thirty students conducted a pet ownership survey.Results of the survey indicate that 8 students own a cat,15 students own a dog,and 5 students own both a cat and a dog.How many of the students surveyed own only a dog?

A)10
B)22
C)15
D)18

Question 60

A combination lock on a suitcase has 5 wheels,each labeled with digits 1 to 8.How many 5-digit combination lock codes are possible if no digit can be repeated?

Question 61

Solve the problem.
A software company employs 9 sales representatives and 8 technical representatives.How many ways can the company select 5 of these employees to send to a computer convention if at least 4 technical representatives must attend the convention?

A)1440
B)180
C)686
D)360

Question 62

Solve the problem.
In how many ways can a student work 8 out of 10 questions on an exam?

A)45
B)16
C)100,000,000
D)90

Question 63

Determine whether the selection is a permutation, a combination, or neither.
An animal trainer selects 2 of the 5 baboons to showcase on the talk show.

A)permutation
B)combination
C)neither

Question 64

Solve the problem.
If the police have 9 suspects,how many different ways can they select 5 for a lineup?

A)45
B)3024
C)15,120
D)126

Question 65

Solve the problem.
How many 4-digit numbers can be formed using the digits 0,1,2,3,4,5,6,if repetition of digits is not allowed?

A)24
B)2401
C)23
D)840

Question 66

Solve the problem.
How many ways can a committee of 5 be selected from a club with 10 members?

A)252
B)30,240
C)50
D)100,000

Question 67

Solve the problem.
How many ways can a committee of 3 be selected from a club with 12 members?

A)1320
B)110
C)220
D)6

Question 68

Solve the problem.
A home cooking equipment supply company employs 7 sales representatives and 6 product designers.How many ways can this company select 4 of these employees to send to a product demonstration convention in Oahu if at least 3 product designers must attend?

A)168
B)42
C)155
D)504

Question 69

Solve the problem.
In a Power Ball lottery,5 numbers between 1 and 12 inclusive are drawn.These are the winning numbers.How many different selections are possible? Assume that the order in which the numbers are drawn is not important.

A)248,832
B)792
C)95,040
D)120

Question 70

Solve the problem.
A pollster wants to minimize the effect the order of the questions has on a person's response to a survey.How many different surveys are required to cover all possible arrangements if there are 6 questions on the survey?

A)36
B)6
C)720
D)120