Midterm Exam Review
Chapter 1
Set Theory
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. What is the meaning of complement in set theory?
A.
B.
C.
D.
____
2. What is the meaning of disjoint in set theory?
A.
B.
C.
D.
____
N, the set of natural numbers, and I, the set of integers
T, the set of all triangles, and C, the set of all circles
N, the set of natural numbers, and P, the set of positive integers
none of the above
4. Which pair of sets represents one set being a subset of another but is not equal?
A.
B.
C.
D.
____
two or more sets having no elements in common
two or more sets that do not match
sets that are in different universal sets
sets that contain no elements
3. Which pair of sets represents disjoint sets?
A.
B.
C.
D.
____
all the elements in the universal set that are not identical
a set of elements that work well with a given set
all the elements of a universal set that do not belong to a subset of it
all the elements that are the opposite of the elements in a given set
N, the set of natural numbers, and I, the set of integers
T, the set of all triangles, and C, the set of all circles
N, the set of natural numbers, and P, the set of positive integers
none of the above
5. Rahim described the set as follows:
• M = {all of the foods he eats}
• D = {his favourite desserts}
• V = {his favourite vegetables}
• F = {his favourite fruits}
Which are the disjoint sets?
A.
B.
C.
D.
M and D
M and V
M and F
V and F
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Midterm Exam Review
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Chapter 1
Set Theory
6. Rahim described the set as follows:
• M = {all of the foods he eats}
• D = {his favourite desserts}
• V = {his favourite vegetables}
• F = {his favourite fruits}
Assume Rahim likes some fruit for dessert. Which statement is true?
A.
B.
C.
D.
____
The universal set is D, Rahim’s favourite desserts.
Set F is a subset of D.
Set D is a subset of V.
Set M and set V are mutually exclusive.
7. Given the following situation:
• the universal set U = {positive integers less than 20}
• X = {4, 5, 6, 7, 8}
• P = {prime numbers of U}
• O = {odd numbers of U}
Which diagram represents the situation?
A.
C.
B.
D.
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Midterm Exam Review
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Chapter 1
Set Theory
8. There are 28 students in Mr. Connelly’s Grade 12 mathematics class.
The number of students in the yearbook club and the number of students on student council are
shown in the Venn diagram. Use the diagram to answer the following questions.
How many students are in both the yearbook club and on the student council?
A.
B.
C.
D.
____
2
5
1
7
9. There are 28 students in Mr. Connelly’s Grade 12 mathematics class.
The number of students in the yearbook club and the number of students on student council are
shown in the Venn diagram. Use the diagram to answer the following questions.
How many students are in at least one of the yearbook club or on student council?
A.
B.
C.
D.
____
2
5
8
7
10. There are 28 students in Mr. Connelly’s Grade 12 mathematics class.
The number of students in the yearbook club and the number of students on student council are
shown in the Venn diagram. Use the diagram to answer the following questions.
How many students are on the student council but not in the yearbook club?
A.
B.
C.
D.
2
5
1
7
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Midterm Exam Review
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Chapter 1
Set Theory
11. Consider the following Venn diagram of herbivores and carnivores:
Determine n(H ∪ C).
A.
B.
C.
D.
____
2
9
4
3
12. Consider the following Venn diagram of herbivores and carnivores:
Determine H ∩ C.
A.
B.
C.
D.
____
{moose, rabbit, deer, squirrel}
{bear, raccoon, badger}
{cougar, wolf}
{moose, rabbit, deer, squirrel, bear, raccoon, badger, cougar, wolf}
13. Consider the following Venn diagram of foods we eat raw or cooked:
Determine H ∪ C.
A.
B.
C.
D.
{fish, spinach, apples, cucumber, lettuce, chicken, pork, rice, pasta, potatoes}
{chicken, pork, rice, pasta, potatoes}
{cucumber, lettuce}
{fish, spinach, apples}
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Midterm Exam Review
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Chapter 1
Set Theory
14. Consider the following Venn diagram of foods we eat raw or cooked:
Determine n(H ∩ C).
A.
B.
C.
D.
____
____
2
5
11
3
15. Consider the following two sets:
• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
• B = {–9, –6, –3, 0, 3, 6, 9, 12}
Which Venn diagram correctly represents these two sets?
A.
C.
B.
D.
16. Consider the following two sets:
• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
• B = {–9, –6, –3, 0, 3, 6, 9, 12}
Determine n(A ∪ B).
A.
B.
C.
D.
8
11
16
20
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Midterm Exam Review
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{–9, –6, –3, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}
{0, 1, 2, –3, 4, 5, –6, 7, 8, –9, 10, 11, 12}
{1, 2, 4, 5, 7, 8, 10, 11}
{–9, –6, –3, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
18. Consider the following two sets:
• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
• B = {–9, –6, –3, 0, 3, 6, 9, 12}
Determine A ∩ B.
A.
B.
C.
D.
____
Set Theory
17. Consider the following two sets:
• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
• B = {–9, –6, –3, 0, 3, 6, 9, 12}
Determine A ∪ B.
A.
B.
C.
D.
____
Chapter 1
{3, 6, 9, 12}
{0, 3, 6, 9, 12}
{1, 2, 4, 5, 7, 8, 10, 11}
{–9, –6, 6, 9}
19. A restaurant offers Chinese, Thai, and Korean food. The following Venn diagram shows the types
of food the customers like.
Use the diagram to determine n(K \ T).
A.
B.
C.
D.
____
10
19
50
23
20. Some table games use a board, dice, or cards, or a combination these. The following Venn diagram
shows the number of games that use these tools.
Use the diagram to determine the number of games that use exactly two of these tools.
A.
B.
C.
D.
13
51
25
74
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Midterm Exam Review
Chapter 1
Set Theory
Short Answer
1. Tania recorded the 16 possible sums that can occur when you roll two four-sided dice.
• S = {all possible sums}
• L = {all sums less than 4}
• G = {all sums greater than 4}
• F = {all sums equal to 4}
What is n(L)?
2. Tania recorded the 16 possible sums that can occur when you roll two four-sided dice.
• S = {all possible sums}
• L = {all sums less than 4}
• G = {all sums greater than 4}
• F = {all sums equal to 4}
What is n(F)?
3. What is the set notation for the set of all positive real numbers that are less than 22?
4. Consider the following information.
• the universal set U = {all grade 12 courses}
• S = {all social studies courses}
• H = {all history courses}
• G = {all geography courses}
• M = {all mathematics courses}
List the pairs of disjoint sets.
5. Consider the following information.
• the universal set U = {all grade 12 courses}
• S = {all social studies courses}
• H = {all history courses}
• G = {all geography courses}
• M = {all mathematics courses}
What new set could belong to set U but not to set S or set M?
6. Given the following situation:
• the universal set U = {positive integers less than 20}
• X = {4, 5, 6, 7, 8}
• P = {prime numbers of U}
• O = {odd numbers of U}
Determine n(X and P).
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Midterm Exam Review
Chapter 1
Set Theory
7. If you draw a card at random from a standard deck of cards, you will draw a card that is either red
(R) or black (B). The card will also either be a number card (N) or a face card (F). Determine n(B
∩ N).
8. A music school offers lessons on 8 different instruments.
piano
bagpipe
recorder
harmonica
violin
xylophone
steel drum
banjo
Determine the number of instruments with neither keys nor strings, (K ∪ S)´.
9. A music school offers lessons on 8 different instruments.
piano
bagpipe
recorder
guitar
violin
xylophone
steel drum
banjo
Determine the number of instruments that have neither a mouthpiece nor keys (M ∪ K)´.
10. Flightless birds include the ostrich, emu, penguin, and kiwi. Arctic birds include the snow goose,
Arctic tern, osprey, penguin, and red-tailed hawk. Determine the union and intersection of these
two sets.
11. Grade 12 students were surveyed about their extra curricular activities.
• 58% belonged to a sports team (S)
• 63% belonged to a band or choir (B)
• 47% belonged to a school club (C)
• 24% belonged to a sports team and a band or choir
• 21% belonged to a sports team and a school club
• 36% belonged to a band or choir and a school club
• 19% engaged in all three activities
What percent of students are involved in only one of these activities? Write your answer in set
notation.
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Midterm Exam Review
Chapter 1
Set Theory
12. The city surveyed 3000 people about how they travel to work.
• 1978 took public transit (P)
• 1494 drove (D)
• 818 cycled (C)
• 731 took public transit and drove only
• 298 took public transit and cycled only
• 27 drove and cycled only
• 164 used all three modes of transportation
How many people travel to work some other way? Use a Venn diagram to show your answer.
13. The city surveyed 3000 people about how they travel to work.
• 1978 took public transit (P)
• 1494 drove (D)
• 818 cycled (C)
• 731 took public transit and drove only
• 298 took public transit and cycled only
• 27 drove and cycled only
• 164 used all three modes of transportation
How many people use public transit only? Use a Venn diagram to show your answer.
14. The city surveyed 3000 people about how they travel to work.
• 1978 took public transit (P)
• 1494 drove (D)
• 818 cycled (C)
• 731 took public transit and drove only
• 298 took public transit and cycled only
• 27 drove and cycled only
• 164 used all three modes of transportation
How many people did not cycle or use public transit? Write your answer using set notation.
15. The city surveyed 3000 people about how they travel to work.
• 1978 took public transit (P)
• 1494 drove (D)
• 818 cycled (C)
• 731 took public transit and drove only
• 298 took public transit and cycled only
• 27 drove and cycled only
• 164 used all three modes of transportation
How many people used only one mode of transport? Write your answer using set notation.
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Midterm Exam Review
Chapter 1
Set Theory
Problems
1. a) Indicate the multiples of 2 and 3, from 1 to 100, using set notation. List any subsets.
b) Represent the sets and subsets in a Venn diagram.
2. a) Determine how many natural numbers from 1 to 100 are
i) odd, square numbers
ii) even, square numbers
iii) not square numbers
b) How many numbers are square?
3. Paul asked 30 people who saw a movie based on a popular book if they liked the book or the
movie.
• 3 people did not like the movie or the book.
• 15 people liked the movie.
• 22 people liked the book.
Determine how many people liked both the movie and the book, how many liked only the movie,
and how many liked only the book.
4. A game store polled 150 customers about whether they preferred to play strategy games or games
of chance.
• 75 people liked to play both.
• 20 people did not like games.
• 31 people liked only strategy games.
Determine how many people liked only games of chance.
5. Noreen asked 65 people at a gym if they liked cycling or running.
• 7 people did not like either activity.
• 24 people liked cycling.
• 40 people liked running.
Determine how many people liked cycling and running.
6. A pet store polled 500 customers about whether they preferred cats or dogs.
• 273 people liked cats.
• 264 people liked dogs.
• 9 people did not like either cats or dogs.
Determine how many people liked cats and dogs.
7. Liam asked 90 people if they preferred tea or coffee.
• 8 people liked both.
• 55 people liked coffee.
• 32 people liked tea.
Determine how many people did not like coffee or tea. Draw a Venn diagram to show your
solution.
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Midterm Exam Review
Chapter 1
Set Theory
8. A restaurant survey asked 300 people if they preferred Indian or Chinese food.
• 146 people liked both.
• 213 people liked Indian food.
• 219 people liked Chinese food.
Determine how many people did not like Indian or Chinese food. Draw a Venn diagram to show
your solution.
9. The card game Uno has cards divided into 4 colours: red, blue, green, and yellow. For each colour,
there are 19 number cards and 6 action cards. There are also 8 special black action cards.
Determine the following amounts.
a) n(D), the total number of cards in the deck
b) n(A), the total number of action cards in the deck
c) n(G), the total number of green cards in the deck
d) n(N), the total number of number cards in the deck
e) n(A ∪ G)
f ) n(G ∩ N)
10. Martin and Louise want to travel from their home in Vancouver to Prince Edward Island.
a) List the types of transportation they could use on their trip.
b) Chose two ways they could travel to PEI. List the subcategories they could choose from for
each method.
c) Draw a Venn diagram to represent one way they could travel to PEI.
d) What are two other things they should consider when choosing how they will travel to PEI?
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Midterm Exam Review
Chapter 1
Set Theory
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