Chapter 3 Sets
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Transcript of 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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The intersection of two sets, X and Y, can occur inthe following way
X
Y
( X Y ) = Y
Y X
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The intersection of two sets, X and Y, can occur inthe following way
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 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
A B
C
A B C = A C
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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 = C
A
B
C A B C = C
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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
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
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
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A
B
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
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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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Shade the region representing each of the following sets
(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
Shade the region representing each of the following sets
(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