chapter 1 set theory. chapter 1 1.1 types of sets and set notation

Chapter 1SET THEORY

Chapter 11.1 – TYPES OF SETS AND SET NOTATION

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 = { } = ØWe 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 U

The set A’, is the complement of A, and can be defined as:A’ = {3}

EXAMPLEa) 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: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 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}

T is a subset of both F and S. We can write:T F

S

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.

EXAMPLEBilyana 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.

Independent practice

PG. 14-18, #1, 2, 4, 5, 8, 9, 16.

Chapter 11.2 – EXPLORING

RELATIONSHIPS BETWEEN SETS

VENN DIAGRAMSIn 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.

Independent PracticePG. 20-21, #1-5

Chapter 11.3 – INTERSECTION AND UNION OF TWO

SETS

INTERSECTIONS AND UNIONS

Intersections Unions

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…

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) Describe sets C, S, H, and D, and the universal set U for this situation.b) Determine n(C), n(S), n(H), n(D), and n(U).c) Determine the union of S and H. Determine n(S H). d) Describe the intersection of S and H. Determine n(S H).e) 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.

a) Describe sets C, S, H, and D, and the universal set U for this situation.

U = {drawing any of the 52 cards}S = {drawing a spade}H = {drawing a heart}C = {drawing a club}D = {drawing a diamond}b) Determine n(C), n(S), n(H), n(D),

and n(U).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(U) = 52n(C) = 13n(S) = 13n(H) = 13n(D) = 13

c) Determine the union of S and H. Determine n(S H).

S H = {13 spades and 13 hearts}

n(S H) = 26d) Describe the intersection of S and H. Determine n(S H). S H = { } They are mutually exclusive.

n(S H) = 0f) Describe the complement of S H.(S H)’ = {the set of all cards that are

not spades or hearts} (S H)’ = (C D)

NUMBER OF ELEMENTS IN A UNION

Petra thinks that n(S) + n(H) = n(S H). Is she correct?

WORKSHEET

The athletics department at a large high school offers 16 different sports:

Badminton Hockey TennisBasketball LacrosseSoccer Cross-country running Rugby VolleyballCurling Cross-country skiing WrestlingFootball Ultimate FrisbeeGolf Softball

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}

Independent Practice

PG. 32-35, #1, 3, 5, 6, 7, 9, 11, 13.

Chapter 11.4 – APPLICATIONS OF SET THEORY

Complete the worksheet to the best of your abilities, and make sure to use set notation.

EXAMPLE

• 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

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?

a) P = {children with pets} B = {children with a bird} C = {children with a cat} D = {children with dogs}

PB

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

PB

C

D

43

2

Let x be the intersection of all three sets. x = n(B C D)

From the information we know that:n(B C) = 2 + xn(B D) = 3 + x n(C D) = 4 + x

What’s the union of all three sets? n(B C D) = 2813 + 13 + 13 – (2 + x) – (3 + x) – (4 + x)

+ x = 28

30 – 2x = 28 –2x = –2 x = 1

So, one child has all three pets.

x

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• 100 students cycle & use public

transit• 80 students walk and use public

transit• 35 students walk and cycle

• 20 students walk, cycle & use public

• 445 students cycle or use public transit

• 265 student walk or cycleHow many students use green transportation for travelling to and from school?

UT

W

C

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?

20

Can we fill in the rest of the diagram?

Independent Practice

PG. 51-54, #2, 4, 7, 9, 10, 12