chapter 3 sets

8/3/2019 Chapter 3 Sets

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Chapter 3Created By: Mohd Said B Tegoh


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Skills Practice

Sort the objects below into the following groups: fruits, mammals, furniture,metals, fish and birds.

Copper Carp Gold Coconut Tin Eagle Whale Owl Guava

Bear Table Bed Chair Eel Papaya Bat Duck Shark

FRUITS

MAMMALS

FURNITURE

METALS

FISH

BIRDS

Coconut

Grouping things according to their common properties

Guava Papaya

Whale BatBear

Table Bed Chair

Copper Gold Tin

Carp Eel Shark

Eagle DuckOwl


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Sets

A set is a collection or group of objects orthings which have a certain property incommon (specific characteristics). The

objects or things are called the elements ormembers of the set.

A set must be clearly defined so that we can

determine if an object is a member of the setor not.

3.1


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A Describing Sets

A set can be defined in two ways

a) Description

b) Set notation with braces { }

Example

Describing in words:the set of states in Malaysia whose names begin with

the letter S

Selangor, Sarawak, Sabah


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A Describing Sets

Example Selangor, Sarawak, SabahUsing set notation with braces { }

Statement

A = { states in Malaysia whose names begin with the letter S }

Listing the elements within braces

A = { Selangor, Sarawak, Sabah }


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A Describing Sets

Example Selangor, Sarawak, SabahUsing set notation with braces { }

Stating a variable within braces( characteristics of element )

A = { x : x is a state in Malaysia whose first letter is S }

We can denote a set with capital letters as shown in the

example above

A is the set of elements x where x is a state in Malaysiawhose first letter is S


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Describing in words:

the set of multiples of 6 that are less than 72

Using set notation with braces { }

Statement

S = { Multiples of 6 that are less than 72 }

Listing the elements within braces

S = { 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 }

A Describing Sets

Example


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Stating a variable within braces( characteristics of element )

S = { x : x is multiple of 6 and x < 72 }

We can denote a set with capital letters as shown in

the example above

A Describing Sets

Example


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Example 1

List the elements of each of the following sets.

(a) A = { factors of 18 }

(b) B = { consonants in the word MEMBERS }

(c) C = { multiples of 7 which are less than 50 }

Solution

(a) A = { 1,2,3,6,9,18 }

(b) B = { M,B,R,S } The letter M is listed only once

(c) C = { 7,14,21,28,35,42,49 }


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Describe the elements of the following sets.

(a) A = { 2,3,5,7 }(b) B = { 1,3,7,21 }

(c) C = { January, June, July }

(a) A is the set of prime numbers which are less than 10.

(b) B is the set of factors of 21.

(c) C is the set of months of the year whose names begin

with the letter J.

Example 1

Solution


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B Identifying The Elements of A Set andUsing The Symbol (epsilon)

The symbol is used to denote is an element of and is

used to represent the membership of an element in a set

The symbol is used to denote not element of

For Example:

T = { Days of the week that start letterT }

Thus,

Tuesday T and Thursday TBut,

Wednesday T and Sunday T


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C Representing Sets Using Venn Diagram

Besides representing sets in words and using braces,

we can also represents set by using a Venn diagram

Usually, shapes such as circles, ovals, rectangles and

triangles are drawn to represent sets in a Venn diagram

In a Venn diagram, the elements of a set can be represented

by a dot

For example; L = { a, b, c, d }L

. a . b

. c . dEach dot in a Venn diagram

represents one element


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D Stating The Number Of Elements In A Set

The number of elements in a set A is represented byn (A)

For example,

(a) A = { a, b, c, d, e }

n (A) = 5

(b) Given that A = { factors of 18 }. Find the value of n (A)

Solution

A = { 1,2,3,6,9,18 }

n (A) = 6


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E Empty Sets

An empty set or a null set is a set that does notcontain any elements

It is denoted by the symbol or { }

For example,

X = { Pupils in your class who are over 35 years old }

X is an empty set, and can be written asX = or X = { }

Note that { 0 } is not empty set. This is because { 0 } contains

one element, 0


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F Equal Sets

A Set A and Set B are equalif they have exactly thesame elements.

That is, the element ofA is an element ofB and element ofB is an element ofA.

For example,

A = { 1,3,5,7 } and B = { 7,5,1,3 }

Sets A and B have the same number of elements and theelements are exactly the same. Thus, A = B


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F Equal Sets

If Set A and Set B do not contain exactly the same elementsor same number of elements, we say that they are not equal.

This is denoted by A B.

For example,

A = { 1,3,5,7 } and B = { 7,5,1,3, 9 }

Sets A and B do not contain exactly the same elements.

Thus, A B.


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3.2 Subsets, Universal Sets and Complementof A Set

Let A = { a, e, i, o, e } and B = { a, e }.

A Subsets

We find that each member of set B is also a member of set A,

that is set B is a subset of set A.

B

The symbol for is a subset of is..

Hence,

A

Set A is a subset of set B if every element of set Ais also an element of set B


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EXAMPLE 1

Given P = { even numbers between 1 and 9 }, Q = { 2,4,8 }and R = { 4,8,12 }, Determine whether each of the followingis true or false

(a) Q P (b) R P (c) R Q

SOLUTION

a

b

c

TrueFalse

False

Each member of set Q is also a member

of set P

Set P does not contain the element 12

Set Q does not contain the element 12


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The relationship between a set and its subset can be shown

using a Venn diagram

For example,Given C = { a, b, c } and D = { a, b, c, d, e, f }.

C D can be represented on a Venn diagram as shown

D

C

.d

.e

.f

Note that set C is containedinside set D.a

.b.c


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3.2Subsets, Universal Sets and Complement

of A Set

When listing the subset of a set , note that

B Listing The Possible Subsets Of A ParticularSet

The number of possible subsets for a certain set A can be

found by using the following formula:

(a) A set is a subset of itself.(b) an empty set ( ) is a subset of every set

Number of subsets = 2n(A),Where n(A) = 3


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BListing The Possible Subsets Of A

Particular Set

Given that the set A = { 7, 8, 9 }

n (A) = 3Number of subsets forA = 2n(A)

= 23

= 8

EXAMPLE

The subset ofA are , {7}, {8}, {9}, {7,8}, {7,9},

{8,9}, {7,8,9}.


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C Universal Sets

A universal set is a set that contains all the elements in

a discussion. It is denoted by the symbol .

F

or ex

ample, = { Positive integers that are less than 10 }

M = { x : x is a multiple of 5 and 0 < x < 10 }

= { 5 }

P = { x : x is a perfect square and 0 < x < 10 }= { 1, 4, 9 }

Thus, the universal set = { 1,2,3,4,5,6,7,8,9 }


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For example, = { Positive integers that are less than 10 }

M = { x : x is a multiple of 5 and 0 < x < 10 }= { 5 }

P = { x : x is a perfect square and 0 < x < 10 }= { 1, 4, 9 }

Thus, the universal set = { 1,2,3,4,5,6,7,8,9 }

The relationship between sets M and P and the universal set can be represented on a Venn diagram as shown

.1.4

.9.5

.2 .3

.6

.7

.8

P M In Venn diagrams,a universal set isusually represented

by rectangle


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D Complement Of A Set

The complement of a set A is the set of elements thatare members of the universal set but not members of

the set A. It is denoted by A

In the Venn diagram as shown,the elements that are not

members ofA are 2,4,6 and 8.Thus, the complement of set A,A = { 2,4,6,8 }

.4

.2 .6

.8

A

.1

.9

.5.3

.7


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D Complement Of A Set

Q

The Venn diagram below shows the relationship Q, Q andthe universal set,

Q

The shaded region portion

outside Q is Q, thecomplement ofQ


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Q

P

Q

COMPLE

MEN

T OF

A SE

T


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3.3 Set Operations

AIntersection Of Two Sets

The intersection of two sets, A and B, is the set where all

its elements are common to sets A and B.

This is denoted by A B.

For example,

A = { c, d, e, f, g, h } and B = { e, f, g, h, i, j }

Thus,

A B = { e, f, g, h } since e, f, g, h are common

to bothA

and B.


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For example,A = { c, d, e, f, g, h } and B = { e, f, g, h, i, j }

This relationship can be illustrated by a Venn diagram as shown

. e. f. g. h

. i

. j. c

. d

A

B

Shaded region

representsthe set A B

Thus, A B = { e, f, g, h }


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The intersection of two sets, X and Y, can occur inthe following way

XY

( X Y ) X

( X Y ) Y


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X

Y

( X Y ) = Y

Y X


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X Y

( X Y ) =


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3.3 SET OPERATIONS

BIntersection Of Three Sets

The intersection of three sets, A, B and C is the set

where all its elements are common to sets A,B and C.

T

his is denoted byA

B C.

For example,

P = { 1,2,3,4 } and Q = { 2,4,6,8 } and R = { 3,4,5,6 }Thus,

P Q R = { 4 } since 4 are common to P, Q and R


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P = { 1,2,3,4 } and Q = { 2,4,6,8 } and R = { 3,4,5,6 }

Thus,

P Q R = { 4 } since 4 are common to P, Q and R

This relationship can be illustrated by a Venn diagramas shown

P R

Q

.1.3

.2

.5

.6

.8

.4

Shaded region

represents the set

P Q R


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The following Venn diagram shows the relationship between

set A, set B, and set C and the intersection of the three sets in

the case ofA

B C.A C

B

Shaded regionrepresents theSet A B C


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the case ofA

B C =A

C

A B

C

A B C = A C


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the case ofA

B C = C

A

B

C A B C = C


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the case ofA B C =

A

BC

A B C =


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3.3 Set Operations

CComplement Of The Intersection Of Two

Sets

The complement of the set A B is the set of

elements that are members of the universal set but not members of the set A B.It is denoted by ( A B )


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The complement of the set A B is the set of elements

that are members of the universal set but not members

of the setA

B. It is denoted by (A

B )

The following diagram shows the region occupied

by the set ( A B )

BA


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3.3 Set Operations

E Union Of Two Sets The union of two sets, A and B, is the set where all its

elements are in set A , or in set B or in both sets A and B.This denoted by A U B.

For example,

A = { k, l, m, n } and B = { n, p, q }

Thus, A U B = { k, l, m, n, p, q }

Note that n written only once


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For example,A = { k, l, m, n } and B = { n, p, q }T

hus, A U B = { k, l, m, n, p, q }

The relationship can be illustrated by a Venn diagram

as shown below.

A B

.k

.l

.m

.n.p

.q

Shaded regionrepresents the

Set A U B


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The following Venn diagrams show the relationship between

set A, set B, and the possible unions of the two sets A U B in

some cases

A B


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BA



some cases


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A

B



some cases


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3.3 Set Operations

F Union Of Three Sets

The union of three sets, A, B and C is the set where all its

elements are in either one of the sets, or in two of the sets

or in all of the three sets. This is denoted by A U B U C.

For example,

A= { 1,2,3,4 }, B = { 2,4,6,8 } and C = { 3,4,5,6 }

Thus, A U B U C = { 1,2,3,4,5,6,8 }


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For example,A = { 1,2,3,4 }, B = { 2,4,6,8 } and C = { 3,4,5,6 }

Thus, A U B U C = { 1,2,3,4,5,6,8 }

This relationship can be illustrated by a Venn diagram as shown

A B

.1

.3

Shaded regionrepresents the

set A U B U C.4

.5

C

.8

.6

.2


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3.3

G Complement Of The Union Of Two Sets

The complement of the set A U B is the set of elements

that are members of the universal set , but not members

of the set A U B. It is denoted by ( A U B ).

Set Operations


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The following diagram shows the region occupied by the set

( A U B ) in a Venn diagram.

A B

Shaded region

represents theSet (A U B)


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3.3

HCombined Operations On A Set

To do combined Operations on a set, do the operations

in brackets first. Then, do the operations from left to right.

Shade the region representing each of the following sets

(a)P Q R

(b)A B

C

Q ( P U R ) ( A B ) U C

Set Operations


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(a) PQ

R

Q ( P U R )

(a) PQ R P Q R

Q ( P U R )( P U R )


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(b) A BC


(b) A BC

A B

C

( A B ) U C

( A B ) U C( A B )


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The Venn diagram in the answer space shows sets P, Q and R. Given that = P U Q U R, on the diagram provided in the answer space, shade

(a) the set P R. (b) the set (P U R) Q. [3 marks]

Answer : (a) P Q

R

(b)

PQ

R


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Answer : (a) PQ

P R

R


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Answer : (b)

(P U R) Q

PQ

R


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The Venn diagram in the answer space shows sets P, Q and R. Given that

= P U Q U R, on the diagram provided in the answer space, shade

(a) the set Q R. (b) the set (P Q) U R. [3 marks]

Answer : (a) P Q

R

(b)

PQ

R


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Answer : (a) PQ

R

(b)

PR

Q R

(P Q) U R

Q


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Theimage cannotbe displayed.Your computer may nothaveenough memory toopen theimage,or the imagemay havebeen corrupted. Restartyour computer,and then open thefileagain.If thered x stillappears,you may havetodeletethe imageand then insertit again.SETS


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The Venn diagram shows the universal set

= { Year Six pupils }Set F = { pupils who read novels }and

Set G = { pupils who read comics }

F G

Given that n (F) = 90, n (G) = 111,

n (F G) = 21 and the number ofpupils who do not read novels or comics is 5,

find the total number of Year Six pupils.


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The Venn diagram shows the universal set

= { Year Six pupils }Set F = { pupils who read novels }and

Set G = { pupils who read comics }

F GGiven that n (F) = 90, n (G) = 111,

n (F G) = 21 and the number ofpupils who do not read novels or comics is 5,

find the total number of Year Six pupils.

SOLUTION

n (F G) = 21

Pupils who do not read novels or comics = 5

5

69 90

Total number of Year Six Pupils

= 69 + 21 90+ + 5 = 185

21


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The following information was obtained interview involving 52 students:

30 could answerQuestionA.28 could answerQuestion B.

40 could answerQuestion C.12 could answerQuestionA and B.

19 could answerQuestion C and A.

11 could answer all three questions.There was no student who could not answer all three questions.

(a) Draw a Venn diagram to display the above information.

(b) How many could answer

(i) only question A ?

(ii) only question B ?(iii) only question C ?(iv) question A and B but not C ?

(v) question B and C but not A ?(vi) question C and A but not B ?


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SOLUTION

A B

C

10

x

40 -19 - x

30 + (21 x) + x + (16 x) = 52n (A) = 30

n (A B) = 12

n (B) = 28

n (C) = 40

n (C A) = 19

n (A B C) = 11

30 + 21 x + x + 16 x = 52

67 x = 52

x = 15

15

6

11

11

8

A B


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A B

C

11

110

15

6

1

SOLUTION

(i) Number of students who could

answer only questionA

= 10(ii) Number of students who could

answer only question B = 1

(iii) Number of students who couldanswer only question C = 6

(iv) Number of students who couldanswer question A and B but not C

= 1

(v) Number of students who could

answer question B and C but not A= 15

(vi) Number of students who could

answer question C and A but not B= 8

8


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integer}anis,3119:{ xxx ee!\It is given that the universal set,and set

Find set R' .

A {19, 23, 29, 31}

B {21, 23, 25, 27, 29}

C {20, 21, 23, 25, 29, 30}

D {19, 22, 24, 26, 27, 28, 31}

R = { x: x is a number such that the sum of its two

digits is a prime number }


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integer}anis,3119:{ xxx ee!\

!\ { 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,

30, 31 }

R = { x: x is a number such that the sum of its

two digits is a prime number }

R = { 20, 21, 23, 25, 29, 30}

R ={ 19, 22, 24, 26, 27, 28, 31}

2 3 5 7 11 3

SOLUTION


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integer},anis,189{ xx ee!\ }14:{ "! xxE

xxF :{! FE

Given that ,

and

A 2

B 4

C 6

D 8

!\ { 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 }

{ 15, 16, 17, 18 }E =

{ 9, 12, 15, 18 }F =

{15, 18}E F =

n(E F) = 2

is a multiple of 3}, find n( ).


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WVU !\

2

U

V

76

41

3

5W

9

DIAGRAM 15

The Venn diagram in Diagram 15 shows the set U, set V

and set W.Given that

List all the elements of the set UWV

A { 2, 4, 9 }

B { 2, 3, 4, 7 }

C { 1, 5, 6, 9 }

D{ 1, 2, 3, 4, 5, 6, 7 }


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2

U

V

76

41

3

5

W

9

{2, 4, 9}

U W VSOLUTION

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