SET THEORY
Chapter 1
1.1 – TYPES OF SETS
AND SET NOTATION
Chapter 1
SET THEORY
We use sets to help us organize things
into categories. A set is a collection of
distinguishable objects; for instance,
the set of whole numbers is:
W = {0, 1, 2, 3, …}
An element is an object/number/word
in a set. The universal set is a set of all
elements under consideration for a
particular context.
List the elements of the universal set of Canadian provinces and territories, C.
 C = {Yukon, British Columbia, Northwest Territories, Alberta,
Saskatchewan, Nunavut, Manitoba, Ontario, Quebec, New Brunswick,
Newfoundland and Labrador, Nova Scotia, Prince Edward Island}
A subset is a set whose elements all belong to another set. What are some subsets of
the set C?
SET THEORY
Consider the subset of the Western
provinces and territories, W. What
might W look like?
 W = {British Columbia, Yukon,
Northwest Territories, Alberta,
Saskatchewan}
We can use Venn diagrams to
represent sets and subsets. What will
the Venn Diagram for C and W look
like?
Consider the set that is opposite of W. Let’s call it W’.
What would that set look like?
 W’ = {Nunavut, Manitoba, Ontario, Quebec,
Newfoundland, Prince Edward Island, New
Brunswick, Nova Scotia}
We call this set the complement of W.
SET THEORY
Consider the set of Canadian
provinces south of Mexico, M.
M={}=Ø
We call this the empty set.
Disjoint sets are two or more sets
that have no elements in common.
What’s an example two subsets of C
that are a pair of disjoint sets?
Consider the subsets T (the set of
territories) and P (the set of provinces).
What would the Venn diagram look like for
C, W, T, and P?
NOTATION
These are some of the symbols and notation we need to know about sets:
Sets are defined using brackets. For example, to define the universal set of the
numbers 1, 2, and 3, list its elements:
U = {1, 2, 3}
Consider the set A = {1, 2}. All elements of A are also elements of U, so A is a
subset of U:
A ÌU
The set A’, is the complement of A, and can be defined as:
A’ = {3}
EXAMPLE
a) Indicate the multiples of 5 and 10, from 1 to 500, using set notation. List any
subsets.
b) Represent the sets and subsets in a Venn diagram.
a) Consider the set S, of all natural numbers between 1 to 500.
S = {1, 2, 3, … , 498, 499, 500}
Another way to write this is:
T is a subset
S = {x | 1 ≤ x ≤ 500, x E N}
Now, consider set F, the set of multiples of 5 from 1 to 500.
F = {5, 10, 15, … , 490, 495, 500}
Another way to write this is:
F = {f | f = 5x, 1 ≤ x ≤ 100, x E N}
F is a subset of S, so we write F Ì S.
Consider set T, the set of multiples of 10 from 1 to 500.
T = {10, 20, 30, … , 480, 490, 500}
Another way to write this is:
T = {t | t = 10x, 1 ≤ x ≤ 50, x E N}
of
both F and S.
We can write:
T Ì F ÌS
Alden and Connie rescue homeless animals and advertise in the local newspaper to find
homes for the animals. They are setting up a web page to help them advertise the
animals that are available. They currently have dogs, cats, rabbits, ferrets, parrots,
lovebirds, macaws, iguanas, and snakes.
a) Design a way to organize the animals on the web page. Represent your organization
using a Venn diagram.
b) Name any disjoint sets and show which sets are subsets of one another.
c) Alden said that the set of fur-bearing animals could form one subset. Name another
set of animals that is equal to this subset.
EXAMPLE
Bilyana recorded the possible sums that can occur
when you roll two four-sided dice in an outcome table.
a) Display the following sets in one Venn diagram:
• rolls that a produce a sum less than 5
• rolls that produce a sum greater than 5
b) Record the number of elements in each set.
a)
S = {all possible sums}
L = {all sums less than 5}
G = {all sums greater than 5}
 L = {(1, 1), (2, 1), (3, 1), (1, 2),
(2, 2), (1, 3)}
 G = {(4, 2), (3, 3), (4, 3), (2, 4),
(3, 4), (4, 4)}
These sets are disjoint. We call this
type of event mutually exclusive,
because they cannot happen at the
same time.
PG. 14-18, #1, 2, 4, 5, 8,
9, 16.
Independent
practice
1.2 – EXPLORING
RELATIONSHIPS
BETWEEN SETS
Chapter 1
VENN DIAGRAMS
In an Alberta school, there are 65 Grade 12 students. Of these students, 23 play
volleyball and 26 play basketball. There are 31 students who do not play either sport.
The following Venn diagram represents the sets of students.
PG. 20-21, #1-5
Independent
Practice
1.3 – INTERSECTION
AND UNION OF TWO
SETS
Chapter 1
INTERSECTIONS AND UNIONS
Intersections
Unions
AÇB
AÈB
The intersection of A and B
includes all of the elements
that are common to both set A
and set B. (All of the elements
that are in both sets).
The union of A and B includes
all of the elements that are in
either A or B.
A\B
A\B is read as “A minus B.” It includes the set of elements that are in set A but not in
set B.
What will the Venn diagram of A\B look like when…
BÌ A
A and B are disjoint
A and B intersect
If you draw a card at random from a standard deck of cards, you will draw a card from
one of four suits: clubs (C), spades (S), hearts (H), or diamonds (D).
a)
b)
c)
d)
e)
Describe sets C, S, H, and D, and the universal set U for this situation.
Determine n(C), n(S), n(H), n(D), and n(U).
Determine the union of S and H. Determine n(S H).
Describe the intersection of S and H. Determine n(S Ç H).
Determine whether the events that are described by sets S and H are mutually
exclusive, and whether sets S and H are disjoint.
f) Describe the complement of S
H.
È
È
a) Describe sets C, S, H, and D, and the universal set U for this situation.
U = {drawing any of the 52 cards}
c) Determine the union of S and H.
Determine n(S H).
È
S = {drawing a spade}
H = {drawing a heart}
S
C = {drawing a club}
n(S
D = {drawing a diamond}
b) Determine n(C), n(S), n(H), n(D), and n(U).
S ÇH={}
 They are mutually exclusive.
n(S Ç H) = 0
n(U) = 52
n(S) = 13
n(H) = 13
n(D) = 13
È H) = 26
d) Describe the intersection of S
and H. Determine n(S Ç H).
The notation n(A) means the number of
elements in set A. So how many
elements are in sets C, S, H, D, and U?
n(C) = 13
È H = {13 spades and 13 hearts}
f) Describe the complement of S
È H.
ÈH)’ = {the set of all cards that are
not spades or hearts}
(S ÈH)’ = (C È D)
(S
NUMBER OF ELEMENTS IN A UNION
Petra thinks that n(S) + n(H) = n(S È H). Is she correct?
WORKSHEET
The athletics department at a large high school offers 16 different sports:
Badminton
Basketball
Cross-country running
Curling
Football
Golf
Hockey
Lacrosse
Rugby
Cross-country skiing
Ultimate Frisbee
Softball
Tennis
Soccer
Volleyball
Wrestling
Make sure to use set notation, including unions,
intersections and n(A) throughout the worksheet.
EXAMPLE
Jamaal surveyed 34 people at his gym. He learned that 16 people do weight training
three times a week, 21 people do cardio training three times a week, and 6 people train
fewer than three times a week. How can Jamaal interpret his results? Draw a Venn
diagram.
Let:
G = {all the people surveyed at the gym}
W = {people who do weight training}
C = {people who do cardio training}
PG. 32-35, #1, 3, 5, 6, 7,
9, 11, 13.
Independent
Practice
1.4 – APPLICATIONS OF
SET THEORY
Chapter 1
Complete the worksheet to the best of
your abilities, and make sure to use
set notation.
EXAMPLE
Use the information to answer these questions:
a) How many children have a cat, a dog, and a bird?
b) How many children have only one pet?
•
•
•
•
28 children have a dog, a cat, or a bird
13 children have a dog
13 children have a cat
13 children have a bird
a) P = {children with pets}
B = {children with a bird}
C = {children with a cat}
D = {children with dogs}
•
•
•
•
4 children have only a dog and a cat
3 children have only a dog and a bird
2 children have only a cat and a bard
No child has two of each type of pet
P
B
C
D
•
•
•
•
28 children have a dog, a cat, or a bird
13 children have a dog
13 children have a cat
13 children have a bird
•
•
•
•
4 children have only a dog and a cat
3 children have only a dog and a bird
2 children have only a cat and a bard
No child has two of each type of pet
Let x be the intersection of all three sets.
 x = n(B Ç C Ç D)
P
B
2
From the information we know that:
n(B ÇC) = 2 + x
n(B ÇD) = 3 + x
n(C ÇD) = 4 + x
What’s the union of all three sets?
 n(B C
D) = 28
È È
13 + 13 + 13 – (2 + x) – (3 + x) – (4 + x) + x = 28
 30 – 2x = 28
 –2x = –2
x=1
So, one child has all
three pets.
3
x
4
C
D
Fill in the rest of the Venn
diagram. How many
children have each pet?
Shannon’s high school starts a campaign to encourage students to use “green”
transportation for travelling to and from school. At the end of the first semester,
Shannon’s class surveys the 750 students in the school to see if the campaign is
working. They obtain these results:
•
•
•
•
370 students use public transit
• 20 students walk, cycle & use public
100 students cycle & use public transit • 445 students cycle or use public transit
80 students walk and use public transit • 265 student walk or cycle
35 students walk and cycle
How many students use green transportation for travelling to and from school?
Let U represent the universal set:
U = {students who attend the school}
T = {students who use public transit}
W = {students who walk}
C = {students who cycle}
How many students use all three
types of transportation?
Can we fill in the rest of the diagram?
U
T
20
C
W
PG. 51-54, #2, 4, 7, 9,
10, 12
Independent
Practice