LOGICAL REASONING (EDITED TXT BOOK)

CHAPTER

3 Logical Reasoning

You will learn
tatements
rguments

The Parliament is the legislative body of the ederal
government ll acts of law must be debated and approved by
the ouse of Representatives and the enate before the approval
of the ang di Pertuan gong

o you now why every act of law must be debated in the
Parliament

Why Study This Chapter?

part from debate, logical reasoning is often applied in the
electrical engineering field especially in electronic circuit t also
plays an important role in computer programming and computer
hardware design n our daily lives, logical reasoning is applied
in the problem solving process because it enhances our critical
and analytical thin ing s ills
54

CHAPTER 3

Walking Through Time

WORD BANK Aristotle
C C
• converse • akas
ccording to ristotle, logic is not a part of
• deductive • deduktif Philosophy but it is ust an instrument used
by philosophers and scientists ristotle
• argument • hujah used logic as a technique for debating and
linguistics nowledge
• implication • implikasi

• inductive • induktif

• contrapositive • kontrapositif

• negation • penafian

• quantifier • pengkuantiti

• statement • pernyataan

• compound statement • pernyataan majmuk

• inverse • songsangan http bt sasbadi com m

55

Chapter 3 Logical Reasoning

3.1 Statements

What does a statement mean and how do you determine Learning
the truth value of a statement? Standard

How many Explain the meaning of
vertices does a statement and hence
a cuboid have? determine the truth value
of a statement.

CHAPTER 3 A cuboid has
12 vertices.

A cuboid has
8 vertices.

ost of the mathematical results involve statements

statement is a sentence of which the truth value can be determined,
that is either true or false, but not both

Question, exclamation and command sentences are not statements The truth values of these
sentences cannot be determined
oth answers given in the above dialogue are statements The statement of cuboid has
vertices is a true statement while the statement of cuboid has vertices is a false statement

tatements can be divided into true statements and false statements

Example 1

etermine whether each sentence below is a statement or Malaysiaku
not a statement ustify your answers

a Please send the wor boo alaysia

b uala Lumpur Tower is the tallest tower in

c ow do you come to school
d x
e Ͻ

Solution: The Kuala Lumpur Tower
a ot a statement because the truth value cannot be determined is located at the peak of
b statement because it is true Bukit Nanas. This
c ot a statement because the truth value cannot be determined tower with a height of
d ot a statement because the truth value cannot be determined 421 metres is the 7th
e statement because it is false tallest communication
tower in the world and the
tallest in Southeast Asia.

56

Chapter 3 Logical Reasoning

Mind Stimulation 1

Aim: To determine the truth values of the given statements

Steps:

1. ivide the class into groups
2. tate whether the mathematical sentences provided in the ctivity heet are true

statements or false statements with Round Robin
3. iscuss and draw conclusions from the findings of this activity
Activity Sheet:

Mathematical sentence Truth value CHAPTER 3

a

b 3

c

d 3



e (x y x xy y

f a, b has subsets

g is a factor of

h The lowest common multiple of the numbers and is

Discussion:
re all the mathematical statements true iscuss your reasons

rom the activity in ind timulation , it is found that

ot all the mathematical statements are true The truth values of the
mathematical statements can be determined

Example 2

etermine whether the following statements are true or false f it is false, prove it

a x y x xy y , x ≠ , y ≠ INFO ZONE
b x Ͻ , x ∈ R
c Ͼ x ∈ R means x is an
d ∈ actors of element of the real
e , , ∪ Prime numbers , , numbers.

Solution: eal num er can
be defined as any
a alse Let x and y b alse Ͼ rational numbers or
irrational numbers.
x y c True
, Prime nuSmabiezrssebenar
d True
e alse
x xy y
, , ∪ , , , , ,


ence, x y ≠ x xy y

57

CHAPTER 3 Chapter 3 Logical Reasoning

Example 3
etermine whether the following mathematics statements are true or false Explain if the statement
is false
a ll polygons have diagonals
b ome perfect squares are whole numbers
c ll even numbers have prime factors
d ome straight lines intersect the y axis

Solution:
a alse Triangles do not have diagonals
b alse ll perfect squares are whole numbers
c True
d True

3.1a

1. etermine whether each sentence below is a statement
or not a statement ustify your answers

a Let s play in the field
b alaysia is located in sian continent
c s
d x Ͼ x
e x

2. Construct a true statement by using the given digits
and symbols

a , , , , Ͼ
b , , , , ʚ
c 6, 4, ϫ, 3 ,
d x , р, x
e , , , , ,

3. etermine whether the following statements are true or false
a ll quadrilaterals have right angles
b ome rhombuses have four equal sides
c ll triangles have equal sides
d ome polygons have five sides
e ll circles can be divided equally into eight sectors

58

Chapter 3 Logical Reasoning

How do you negate a statement? Learning
Standard
We use the word “no” or “not” to negate
a statement. The negation of statement p Negate a statement.
is written as ~p.
Indicator
Example 4 ‘~p’ is read as
‘tilde p’.

orm a negation p for each of the following statements p by using the word no or not CHAPTER 3
a is a multiple of
b is a prime number
c ll multiples of are multiples of
d m is equal to mm

Solution: b is not a prime number
a is not a multiple of d m is not equal to mm
c ot all multiples of are multiples of

Mind Stimulation 2

Aim: To determine the truth value of a statement after the negation
Steps:
1. ivide the class into groups
2. etermine the truth values of the following statements

a ll even numbers are divisible by
b ll factors of are factors of
c is a perfect cube number
d of is
e a, b, c is a subset of a, b ∩ b, c

3. orm a negation for each statement in tep by using the word no or not
4. etermine the truth values of the negation statements in tep

Discussion:
hat can you say about the truth values of the statements in tep before and after the

negation

rom the activity in ind timulation , it is found that

The truth values change from true to false or vice versa through the process of negation

59

CHAPTER 3 Chapter 3 Logical Reasoning

3.1b

orm a negation p for each of the following statements p by using the word no or not
T hen, determine the truth values of the negations
1. is a multiple of
2. ite has two axes of symmetry
3. cone has one curved surface
4. Two parallel lines have the same gradient
5. ll quadratic equations have two equal roots

How do you determine the truth value of a compound statement?

compound statement is a combination of two or more statements Learning
by using the word and or or Standard

Example 5 Determine the truth value
of a compound statement.

Combine the following statements, p and q, by using the words

i and, ii or

a p pentagon has two diagonals
q heptagon has four diagonals

b p pyramid has five planes
q pyramid has five vertices

c p is an integer
q is an integer

Solution:
a i pentagon has two diagonals and a heptagon has four diagonals
ii pentagon has two diagonals or a heptagon has four diagonals

b i pyramid has five planes and five vertices
ii pyramid has five planes or five vertices

c i and are integers
ii or is an integer

Example 6

etermine the two statements, p and q, in the following compound statements
a Ͼ and Ͻ
b and are prime numbers
c or 3
d Ͻ or Ͻ

60

Solution: Chapter 3 Logical Reasoning
a p Ͼ
q Ͻ b p is a prime number
q is a prime number
c p
q 3 d p Ͻ
q Ͻ

The word and in a mathematical statement means both
while the word or means one of them or both

ased on the picture on the right, three statements CHAPTER 3
p, q and r are formed

p Rashid is running

q o eong is running

r elinda is running

ou can combine two statements by using the
word and or or

p and q Rashid and o eong are running

p and r Rashid and elinda are running

q or r o eong or elinda is running

rom the three compound statements above, we notice that the statement p and q is true because
both Rashid and o eong are running but p and r is false because not both Rashid and elinda
are running owever, the compound statement q or r is true because part of the statement is true

ence, the truth value of a compound statement can be concluded as shown in the truth table below

p q p and q p or q

True True True True

True alse alse True

alse True alse True

alse alse alse alse

Saiz sebenar

61

Chapter 3 Logical Reasoning

Example 7 y y – 3x = 5
y = 3x
etermine the truth values of the following compound statements
x
a and are greater than

b x Ͻ x and is an odd number

c is a perfect square and is a factor of O

d y x is parallel to y x and the y intercept of the straight line y x is

CHAPTER 3 e or ϫ
f ϫ ϫ 3 or
g The sum of interior angles of a triangle or a quadrilateral is

h or

Solution:

Statement Truth value

a p is greater than alse
q is greater than alse
alse
p and q and are greater than

b p x Ͻ x alse
q is an odd number True
alse
p and q x Ͻ x and is an odd number

cp is a perfect square True
True
q is a factor of True

p and q is a perfect square and is a factor of

d p y x is parallel to y x True
q The y intercept of the straight line y x is alse
alse
p and q y x is parallel to y x and the y intercept of the straight line
y x is

e p alse
q ϫ alse
alse
p or q or ϫ

f p ϫ ϫ 3 True
q alse
True
p or q ϫ ϫ 3 or

g p The sum of interior angles of a triangle is alse
q The sum of interior angles of a quadrilateral is True
True
p or q The sum of interior angles of a triangle or a quadrilateral is

hp True
q True
or
Saiz sebenapr or q True

62

Chapter 3 Logical Reasoning

3.1c CHAPTER 3

1. Combine the following statements p and q by using the words given in the brac ets to form
compound statements

a p is a prime factor of or
q is a prime factor of

b p cone has one vertex and
q cone has one plane

c p rhombus is a parallelogram and
q trape ium is a parallelogram

2. etermine the truth values of the following compound statements

a is a multiple of and a perfect square
b hours minutes and minutes seconds
c The coefficient of x is and
d ʦ , , and , ʚ , ,

e can be expressed as a recurring decimal or less than

f 4 or 4 is a proper fraction
g or is an odd number

h 64 or 3

How do you construct a statement in the form of Learning
an implication? Standard

Teacher, can I play Construct statement in
football on the field? the form of implication
(i) if p, then q
(ii) p if and only if q

If you can finish
answering all the
questions, then

you can play.

Implication “If p, then q”

statement if p, then q is nown as an implication where
• p is denoted as the antecedent
• q is denoted as the consequent

63

Chapter 3 Logical Reasoning

Example 8

orm an implication if p, then q with the given antecedent and consequent

a ntecedent k is divisible by b ntecedent et K is a subset of set L
Consequent k is a multiple of Consequent n(K р n(L

Solution: L
a f k is divisible by , then k is a multiple of K
b f set K is a subset of set L, then n(K р n(L

CHAPTER 3 Example 9

etermine the antecedent and consequent for the following implications if p, then q

a f x is a factor of , then x is a factor of
b f x y Ͼ , then x Ͼ y

Solution: b ntecedent x y Ͼ
Consequent x Ͼ y
a ntecedent x is a factor of
Consequent x is a factor of

Implication “p if and only if q”

part from the implication if p, then q , the implication p if and only if q is also used frequently
in logical reasoning

n implication p if and only if q consists of the following two implications

• if p, then q • if q, then p

Example 10

orm an implication p if and only if q with the following implications
a f k is a prime number, then k has only two factors
f k has only two factors, then k is a prime number
b f y axn b is a linear equation, then n
f n , then y axn b is a linear equation

Solution:
a k is a prime number if and only if k has only two factors
b y axn b is a linear equation if and only if n

Example 11

rite two implications based on the implication p if and only if q given below

a r if and only if r b x Ͻ if and only if x Ͻ

Solution: b mplication f x Ͻ , then x Ͻ
mplication f x Ͻ , then x Ͻ
Saiz seba e nmaprlication f r , then r

mplication f r , then r

64

Chapter 3 Logical Reasoning

3.1d

1. orm an implication if p, then q with the given antecedent and consequent
a ntecedent x
Consequent x4

b ntecedent ax3 bx cx d is a cubic equation
Consequent a ≠

c ntecedent n Ͼ n CHAPTER 3

Consequent n Ͻ

d ntecedent m Ͼ
n

Consequent m Ͼ n

2. etermine the antecedent and consequent for the following implications if p, then q
a f x is an even number, then x is an even number
b f set K φ, then n(K
c f x is a whole number, then x is an even number
d f a straight line AB is a tangent to a circle P, then the straight line AB touches the circle P

at one point only

3. orm an implication p if and only if q with the following implications
a f k is a perfect square, then k is a whole number
f k is a whole number, then k is a perfect square

b f P ʝ Q P, then P ʚ Q
f P ʚ Q, then P ʝ Q P

c f pq , then p q and q p
f p q and q p , then pq

d f k , then k k
f k k , then k

4. rite two implications based on the implication p if and only if q given below

a PQR is a regular polygon if and only if PQ = QR = PR

b m is an improper fraction if and only if m Ͼ n
n

c is the y intercept of a straight line y mx c if and only if c

d f (x ax bx + c has a maximum point if and only if a Ͻ

65

Chapter 3 Logical Reasoning Learning
Standard
How do you construct and compare the truth value
of converse, inverse and contrapositive of Construct and compare
an implication? the truth value of
converse, inverse and
efore comparing the truth value of converse, inverse and contrapositive of
contrapositive of an implication, observe and state the differences an implication.
between four statements in the following example

tatement f li is a prefect, then li is a disciplined person

CHAPTER 3 Converse f li is a disciplined person, then li is a prefect

nverse f li is not a prefect, then li is not a disciplined person

Contrapositive f li is not a disciplined person, then li is not a prefect

n general,

tatement f p, then q

Converse f q, then p

nverse f p, then q

Contrapositive f q, then p

Example 12
rite the converse, inverse and contrapositive of the following implications
a f x is a positive number, then x is greater than
b f p q Ͼ , then p q p q Ͼ
c f x , then x

Solution:

a tatement f x is a positive number, then x is greater than

Converse f x is greater than , then x is a positive number

nverse f x is not a positive number, then x is not greater than

Contrapositive f x is not greater than , then x is not a positive number

b tatement f p q Ͼ , then p q p q Ͼ

Converse f p q p q Ͼ , then p q Ͼ TIPS

nverse f p q ≤ , then p q p q ≤ p is a complement of p
Then, the complement of
Contrapositive f p q p q ≤ , then p q ≤ p q Ͼ is p q р

c tatement f x , then x .

Converse f x , then x

nverse f x ≠ , then x ≠

Contrapositive f x ≠ , then x ≠

fter identifying the converse, inverse and contrapositive of an implication, you will evaluate the
truth of the statements mentioned earlier

66

Chapter 3 Logical Reasoning

Mind Stimulation 3

Aim: To compare the truth values of converse, inverse and contrapositive of an implication

Steps:
1. etermine the truth values for the statements p and q of each implication in the

ctivity heet
2. rite the converse, inverse and contrapositive of an implication if p, then q Then,

determine the truth values of the statements

Activity Sheet Truth CHAPTER 3
value
a p is an even number True
q is divisible by True
True
tatement f is an even number, then is divisible by
Converse True

nverse True
Contrapositive
True
b p is a multiple of
q is a multiple of

tatement
Converse f is a multiple of , then is a multiple of

nverse
Contrapositive

c p The sum of interior angles in pentagon PQRST is
q Pentagon PQRST is a quadrilateral

tatement
Converse

nverse f the sum of interior angles in pentagon PQRST is
not , then pentagon PQRST is not a quadrilateral

Contrapositive

d p x Ͻ
q x Ͼ

tatement
Converse

nverse
Contrapositive f x р , then x у

Saiz sebenar

67

Chapter 3 Logical Reasoning

CHAPTER 3 Discussion:
1. Compare the truth value of a contrapositive and the truth value of an implication

if p, then q
2. hat is the relationship between the converse and inverse of an implication if p, then q
3. hat is the difference of the truth values between the converse and inverse of an implication

if p, then q

rom the activity in ind timulation , it is found that

1. The truth value of contrapositive is the same as the truth value of
an implication if p, then q

2. The converse and inverse are contrapositive to each other
3. The converse and inverse also have the same truth value

ence, you can list the truth values of an implication if p, then q , and its corresponding converse,
inverse and contrapositive in the table below

pq Statement Converse Inverse Contrapositive
If p, then q. If q, then p. If ~p, then ~q. If ~q, then ~p.
True True
True alse True True True True
alse True alse True True alse
alse alse True alse alse True
True True True True

n conclusion,


The truth value of an implication if p, then q is always true except when p is true and q is false
happen at the same time f an antecedent is false, then the implication if p, then q is always
true without depending on the truth value of the consequent

Example 13

etermine the truth values of statement, converse, inverse and contrapositive of the implication
f ϫ , then ϫ
Solution:

Antecedent Consequent Truth value

tatement f ϫ , then ϫ True alse alse

Converse f ϫ , then ϫ alse True True

nverse f ϫ ≠ , then ϫ ≠ alse True True

Saiz sebCeonntararpositive f ϫ ≠ , then ϫ ≠ True alse alse

68

Chapter 3 Logical Reasoning

3.1e CHAPTER 3

1. rite the converse, inverse and contrapositive of the following implications

a f x Ͼ , then x Ͼ
b f k k , then k or k

c f ABCD is a parallelogram, then AB is parallel to CD

2. etermine the truth values of implication, converse, inverse and contrapositive for each of the

following statements

a f and are factors of , then ϫ is

b f is a root of x , then is not a root of x x

c f a rectangle has four axes of symmetry, then the rectangle has four sides

d f ϫ , then ϫ

How do you determine a counter-example to negate Learning
the truth of a particular statement? Standard

or a false statement, at least one counter example can be given Determine a counter -example
to negate the truth of that statement or example, the statement to negate the truth of
a particular statement.
ll polygons have two or more diagonals is false as a triangle
does not have a diagonal The triangle is a counter example to
support the false value

Example 14
etermine the truth value of the following mathematical statements f it is false, give one
counter example to support your answer
a The sum of interior angles of all polygons is
b ome prime numbers are even numbers
c and are the factors of
d or is a multiple of

Solution:
a alse because the sum of interior angles of a pentagon is
b True
c alse because is not a factor of
d True

69

CHAPTER 3 Chapter 3 Logical Reasoning

Example 15
rite the mathematical statement requested in the brac ets for each of the following Then,
determine the truth value of each statement written f it is false, give one counter example to
support your answer
a ∈ , , egation
b ll multiples of are multiples of egation
c f x Ͼ , then x Ͼ Converse
d f x is a root of x3 , then x nverse
e f k Ͼ , then k Ͼ Contrapositive

Solution:
a egation ∉ , , alse because is an element of , ,
b egation ot all multiples of are multiples of alse because all the multiples of are

divisible by
c Converse f x Ͼ , then x Ͼ alse because Ͼ but Ͻ
d nverse f x is not a root of x3 , then x ≠ True
e Contrapositive f k р , then k р alse because Ͻ but Ͼ

3.1f

1. etermine the truth values of the following mathematical statements f it is false, give one
counter example to support your answer

a ll rectangles are squares
b ome perfect squares are divisible by
c or have two factors
d is a multiple of and

2. rite the mathematical statement requested in the brac ets for each of the following Then,
determine the truth value of each statement written f it is false, give one counter example
to support your answer

a 8 8 8 egation
b cuboid has four uniform cross sections egation
c f y x is parallel to y x , then y x and y x have the same gradient

Converse
d f a triangle ABC has a right angle at C, then c a b nverse
e f w Ͻ , then w Ͻ Contrapositive

70

Chapter 3 Logical Reasoning

3.2 Arguments

What does an argument mean? What is the difference Learning
between deductive and inductive arguments? Standard

Explain the meaning of
an argument and
differentiate between
deductive and inductive
arguments.

If you finish doing the No, she will not CHAPTER 3
mathematics homework, then praise me.
Puan Saripah will praise you.

rom the above conversation, what is the conclusion that you can ma e as uhaimi finished all
his mathematics homewor

The process of ma ing a conclusion based on statements is nown as argumentation n argument
can consist of several premises and one conclusion premise is a statement that gives information
before ma ing a conclusion and a conclusion is an outcome or a decision ormally, a simple
argument consists of at least two premises and one conclusion

Mind Stimulation 4

Aim: To differentiate between deductive and inductive arguments

Steps:

1. ivide the class into groups
2. ifferentiate whether the statements below are specific statements or general statements
Circle your answer

Statement Type of statement
a The area of triangle ABC is cm pecific eneral
b ll prisms have uniform cross sections pecific eneral
c ll multiples of end with even digits pecific eneral
d is divisible by pecific eneral
e 3 pecific eneral
f The volume of a cube x3, where x is the edge of the cube pecific eneral
g The height of cylinder P is cm pecific eneral
h ll numbers in base consist digits which are smaller than pecific eneral
i (x x has two roots pecific eneral

The volume of a sphere is 43 r3, where r is the radius of the sphere pecific eneral
Discussion
ustify your answer

71

Chapter 3 Logical Reasoning

rom the activity in ind timulation , it is found that

The specific statements are statements that refer to a particular case,
while the general statements are statements that describe a concept

CHAPTER 3 There are two types of arguments, that are
deductive argument and inductive argument. Try
to justify the inductive argument and deductive
argument through Mind Stimulation 5.

Mind Stimulation 5

Aim: To ustify deductive argument and inductive argument
Steps:
1. ivide the class into groups
2. tudy the arguments in ctivity heet
3. Complete ctivity heet by writing premise and the conclusion of each argument

from ctivity heet ence, determine the type of statement for the premise and
conclusion and circle the specific or general words below

Activity Sheet A

Argument

a The area of a circle is r b ll multiples of are multiples of and
Circle A has a radius of cm multiples of
n conclusion, the area of circle A is a multiple of
is cm ence, is a multiple of and

c d 3
3
3
3 3
4

n conclusion n , where Therefore, the number sequence , ,
n , , , ,
, , can be formulated as n3 ,
where n , , , ,

e ll Class Celi pupils scored in f Tigers are carnivores
the athematics test Lions are carnivores
Saiz sebenar Camelia is a pupil in Class Celi Crocodiles are carnivores
n conclusion, Camelia scored Penguins are carnivores
in the athematics test Therefore, all the animals above
are carnivores

72

Chapter 3 Logical Reasoning

Activity Sheet B Conclusion Type of argument
Premise 1 eductive argument
n conclusion, the area of
a The area of a circle is r circle A is cm

eneral pecific eneral pecific

b eneral pecific eductive argument CHAPTER 3
eneral pecific eneral pecific nductive argument
eneral pecific nductive argument
c eneral pecific eductive argument
eneral pecific eneral pecific nductive argument

d
eneral pecific

e
eneral pecific

f
eneral pecific

Discussion:
ased on the types of arguments given, ustify the deductive argument and inductive argument

rom the activity in ind timulation , it is found that

• eductive argument is a process of ma ing a specific conclusion based on general premises
• nductive argument is a process of ma ing a general conclusion based on specific premises

Example 16

etermine whether each argument below is a deductive argument or an inductive argument

a ll acute angles are less than ngle PQR is an acute angle Thus, angle PQR is less
than

b ll sudo u competition representatives are members of the athematics Club amal is
a sudo u competition representative Thus, amal is a member of the athematics Club

c The sum of exterior angles of a triangle is The sum of exterior angles of a quadrilateral
is The sum of exterior angles of a pentagon is Thus, the sum of exterior angles of
each polygon is

d The sum of the digits in is divisible by The sum of the digits in is divisible by

The sum of the digits in is divisible by Thus, the sum of the digits in multSipaleizs soef benar

is divisible by

73

Chapter 3 Logical Reasoning

Solution: b ll sudo u competition
a representatives are

Premise 1: ll acute angles are less Premise 1: members of the
than eneral athematics Club
eneral

Conclusion: Then, angle PQR is less Conclusion: Then, amal is a
than pecific member of the

CHAPTER 3 athematics Club
pecific

eductive argument eductive argument

c The sum of exterior d The sum of the digits
Premise 1: angles of a triangle is Premise 1: in is divisible by
pecific pecific

Then, the sum of exterior Conclusion: Then, the sum of the
Conclusion: angles of each polygon is digits in multiples of
is divisible by
eneral
eneral

nductive argument nductive argument

3.2a

etermine whether each argument below is a deductive argument or an inductive argument

1. ll factors of are factors of , , and are factors of Thus, , , and are factors
of

2. ϫ 3 , 3 ϫ 4 , 4 ϫ Thus, m ϫ n m n

3. , , , Thus, the number pattern , , , can be expressed as n
n , , ,

4. ll regular polygons have sides of equal length ABCDEFG is a regular polygon Thus,
ABCDEFG has sides of equal length

5. ll multiples of end with the digit is a multiple of Thus, ends with the digit

6. , , , Thus, the number sequence , , , can be
expressed as n n , , ,

7. , , , Thus, the number sequence , , , can be
expressed as n n , , ,

8. ll multiples of are multiples of is a multiple of Thus, is a multiple of

9. ll rational numbers can be written in the fraction form is a rational number Thus, can
be written in the fraction form

10. The supplementary angle of is The supplementary angle of is Thus, the
supplementary angle of θ is θ.

74

How do you determine and justify the validity of Chapter 3 Logical Reasoning
a deductive argument and hence determine whether
the valid argument is sound? Learning
Standard
valid deductive argument can be categorised into three forms
Determine and justify the
Premise 1: Form I Form II Form III validity of a deductive
Premise 2: ll A are B f p, then q f p, then q argument and hence
Conclusion: C is A p is true ot q is true determine whether the
C is B q is true ot p is true valid argument is sound.

deductive argument is said to be valid if all the premises and the conclusion are true CHAPTER 3

Premise 1 : ll actresses are good at dancing
Premise 2 : asmine is an actress
Conclusion : asmine is good at dancing

The above argument is a valid argument lthough we now that Premise is false not all
actresses are good at dancing but this argument is still valid because the argument fulfils orm
as shown in the table above ut the argument above is unsound because Premise is false

Mind Stimulation 6

Aim: To determine and ustify the validity of an argument
Steps:
1. ivide the class into groups
2. Observe the following argument

Premise 1 : ll actresses are good at dancing
Premise 2 : asmine is good at dancing
Conclusion : asmine is an actress

Discussion:
a hy is the above argument not valid
b o all true premises in an argument guarantee the validity of the argument

rom the activity in ind timulation , it is found that

The argument above is not valid because it does not comply with any of the three forms
of valid deductive argument Therefore, a true premise does not guarantee the validity of
an argument

n general,
The validity of an argument is determined based on the forms of argument, and not based on the

truth of the premises or the conclusion

75

Chapter 3 Logical Reasoning When you read the first argument on page 75, you may
doubt that “are all the actresses good at dancing?”. The
Are you sure that answer is possibly no. From experience, we know that
all valid arguments premise 1 is not true. Therefore, apart from the validity
of an argument, we also need to discuss whether the
are sound? argument is sound.

A deductive argument is sound if the argument fulfils the
following two conditions:
(i) the argument is valid, and
(ii) all the premises and the conclusion are true.

CHAPTER 3 17

Are the following arguments valid and sound? If it is not, justify your answer.
(a) Premise 1 : All multiples of 16 are even numbers.

Premise 2 : 64 is a multiple of 16.
Conclusion : 64 is an even number.

(b) Premise 1 : All basketballs are in spherical shape.
Premise 2 : The Earth is in spherical shape.
Conclusion : The Earth is a basketball.

(c) Premise 1 : If w Ͻ 9, then w Ͻ 19.
Premise 2 : 4 Ͻ 9.
Conclusion : 4 Ͻ 19.

(d) Premise 1 : If a ≠ 0, then axn + bx + c is a quadratic expression.
Premise 2 : a ≠ 0.
Conclusion : axn + bx + c is a quadratic expression.

(e) Premise 1 : If k is divisible by 8, then k is divisible by 4.
Premise 2 : 12 is not divisible by 8.
Conclusion : 12 is not divisible by 4.

(f) Premise 1 : If k is an even number, then k + 1 is an odd number.
Premise 2 : 8 + 1 is an odd number.
Conclusion : 8 is an even number.

Solution:
(a) Valid and sound.

(b) Not valid because the conclusion is not formed based on the given premises. Hence, the
argument is not sound.

(c) Valid and sound.

(d) Valid but not sound because Premise 1 and conclusion are not true.

(e) Not valid because the conclusion is not formed based on the given premises. Hence, the
argument is not sound.

(f) Not valid because the conclusion is not formed based on the given premises. Hence, the
argument is not sound.

76

Chapter 3 Logical Reasoning CHAPTER 3

3.2b

re the following arguments valid and sound f it is not, ustify your answer
1. Premise ll multiples of are multiples of
Premise is a multiple of
Conclusion is a multiple of

2. Premise ll squares have right angles
Premise PQRS is a square
Conclusion PQRS has right angles

3. Premise f x Ͻ , then x Ͻ
Premise 4 Ͻ
Conclusion Ͻ

4. Premise f k Ͻ , then k Ͼ
Premise Ͻ
Conclusion Ͼ

5. Premise f x is a factor of , then is divisible by x
Premise is divisible by
Conclusion is a factor of

6. Premise f l is parallel to l , then gradient of l gradient of l
Premise radient of l ≠ radient of l
Conclusion l is not parallel to l

7. Premise ll rhombuses have perpendicular diagonals
Premise PQRS has perpendicular diagonals
Conclusion PQRS is a rhombus

8. Premise f x is an even number, then x is an even number
Premise x is not an even number
Conclusion x is not an even number

9. Premise f k Ͼ , then k Ͼ
Premise k р
Conclusion k р

10. Premise ll cubes are cuboids
Premise Ob ect P is a cube
Conclusion Ob ect P is a cuboid

77

Chapter 3 Logical Reasoning

How do you form a valid deductive argument for Learning
a situation? Standard

Example 18 Form a valid deductive
argument for a situation.

orm a valid deductive argument for each of the following situations

a ll mammals are warm blooded Cats are mammals Cats are warm blooded

b f x is greater than , then x has a positive value is greater than has a positive value

CHAPTER 3 c f x is an odd number, then x is divisible by is not divisible by is not
an odd number

Solution:
a Premise ll mammals are warm blooded
Premise Cats are mammals
Conclusion Cats are warm blooded

b Premise f x is greater than , then x has a positive value
Premise is greater than
Conclusion has a positive value

c Premise f x is an odd number, then x is divisible by
Premise is not divisible by
Conclusion is not an odd number

Example 19

rite a conclusion for each of the following deductive arguments to form a valid and sound
deductive argument
a Premise ll whole numbers are real numbers
Premise is a whole number
Conclusion

b Premise f ax bx + c has real roots, then b ac ജ
Premise x px has real roots
Conclusion

c Premise f a straight line y = mx + c is parallel to the x axis, then m
Premise m ≠
Conclusion

Solution:
a Premise ll whole numbers are real numbers
Premise is a whole number
Conclusion is a real number

78

Chapter 3 Logical Reasoning

b Premise f ax bx + c has real roots, then b ac ജ MY MEMORY
Premise x px has real roots For x px
a , b p and c
Conclusion p ജ

c Premise f a straight line y mx + c is parallel to the x axis,
then m

Premise m ≠

Conclusion The straight line y mx + c is not parallel to the x axis

Example 20 CHAPTER 3

rite the premise for each of the following deductive arguments to form a valid and sound deductive
argument

a Premise is a prime number
Premise has only two factors
Conclusion

b Premise f the annual sales of C company exceeds three millions, then the employees
get a bonus of three months salary

Premise

Conclusion The annual sales of C company do not exceed three millions

c Premise f x k, then k is a root of the equation x
Premise
Conclusion x ≠

Solution:

a Premise ll prime numbers have only two factors
Premise is a prime number
Conclusion has only two factors

b Premise f the annual sales of C company exceeds three millions, then the employees
get a bonus of three months salary
Premise The employees of C company do not get a bonus of three months salary
Conclusion The annual sales of C company do not exceed three millions

c Premise f x k, then k is a root of x
Premise is not a root of x
Conclusion x ≠

3.2c

1. rite the conclusion for each of the following deductive arguments to form a valid and sound
deductive argument
a Premise ll the pupils in manah use digital textboo s
Premise Preevena is a pupil of manah
Conclusion

79

CHAPTER 3 Chapter 3 Logical Reasoning

b Premise f ai eng is the champion of the state level
chess competition, then he will get a cash pri e
of R

Premise ai eng is the champion of the state level
chess competition

Conclusion

c Premise f quadrilateral PQRS is a regular polygon, then
quadrilateral PQRS is a square

Premise Quadrilateral PQRS is not a square
Conclusion

d Premise ll isosceles triangles have one axis of symmetry
Premise ABC is an isosceles triangle
Conclusion

e Premise f m n, then m n
Premise m n
Conclusion

f Premise f m р m , then m у
Premise m Ͻ
Conclusion

2. rite the conclusion for each of the following deductive arguments to form a valid
deductive argument

a Premise ll straight lines with ero gradient are parallel to the x axis
Premise
Conclusion traight line AB is parallel to the x axis

b Premise is a multiple of
Premise is divisible by
Conclusion

c Premise f polygon P is a nonagon, then polygon P has nine vertices
Premise
Conclusion Polygon P has nine vertices

d Premise
Premise x Ͼ
Conclusion x Ͼ

e Premise f it is raining today, then the room temperature is lower than C
Premise
Conclusion t is not raining today

f Premise

Saiz seben arPremise x ≠

Conclusion x ≠

80

Chapter 3 Logical Reasoning

How do you determine and justify the strength of Learning
an inductive argument and hence determine whether Standard
the strong argument is cogent?
Determine and justify the
deductive argument emphasises the validity of the argument while strength of an inductive
an inductive argument emphasises the strength of the argument argument and hence
The strength of an inductive argument is determined based on the determine whether
probability that the conclusion is true, assuming that all premises are the strong argument
true To determine an argument that is cogent or not cogent, it needs is cogent.
to be discussed based on the truth of the premises and its conclusion

Example 21 CHAPTER 3

etermine whether the given arguments are strong or wea ence, determine whether the strong
argument is cogent or not cogent and ustify your answer

a Premise The chairs in the living room are red
Premise The chairs in the dining room are red
Premise The chairs in the study room are red
Premise The chairs in the bedroom are red
Conclusion ll chairs in the house are red

b Premise is a multiple of
Premise is a multiple of
Conclusion ll multiples of are multiples of

c Premise mac erel breathes through its gills
Premise shar breathes through its gills
Conclusion ll fishes breathe through their gills

d Premise ϫ
Premise ϫ
Conclusion ll multiples of end with digit or

e Premise is a prime number
Premise is a prime number
Premise is a prime number
Premise is a prime number
Conclusion ll prime numbers are divisible by and itself

Solution:

a This argument is wea and not cogent because although the premises are true, the conclusion
is probably false

b This argument is strong and cogent because all the premises and conclusion are true

c This argument is wea and not cogent because although the premises are true, the conclusion
is false

d This argument is strong and cogent because all the premises and conclusion are true

e This argument is strong but not cogent because premise is false

81

Chapter 3 Logical Reasoning

rom Example , it is found that the number of premises does not guarantee a strong argument
because the strength of an argument depends on the truth value of the conclusion wea argument
is not cogent while a strong argument will only be cogent if all its premises are true

True conclusion Inductive Argument False conclusion

All premises Strong All premises Weak
are false
are true

Cogent Not Cogent Not Cogent

CHAPTER 3 3.2d

etermine whether the given arguments are strong or wea , and cogent or not cogent ustify
your answers
1. Premise The table is made of wood
Premise The chair is made of wood
Premise The cupboard is made of wood
Conclusion ll furniture are made of wood
2. Premise k k
Premise k8 k
Conclusion k m n k mn
3. Premise 3 is divisible by
Premise is divisible by
Conclusion n is divisible by
4. Premise ϫ
Premise ϫ
Premise ϫ
Conclusion The product of multiples of and end with digit
5. Premise is a multiple of
Premise is a multiple of
Premise is a multiple of
Conclusion ll the multiples of are even numbers
6. Premise Rats have legs
Premise Cats have legs
Premise orses have legs
Conclusion ll animals have legs

82

Chapter 3 Logical Reasoning

How do you form a strong inductive argument of Learning
a certain situation? Standard

strong and cogent inductive argument depends on the true premises Form a strong inductive
and conclusion The given premises are the evidence or support to argument of a certain
the conclusion made nductive reasoning can be carried out in the situation.
following steps

Observe some examples Observe the Make a general
or specific situations. common features. conclusion.

Example 22 CHAPTER 3

orm a strong inductive conclusion for each of the following number sequences

a equence , , , , b equence , , , ,
34
3
4

3



c equence , , , , d equence , , , ,





Solution:
a n n , , , ,
c n n , , , ,
b n n , , , ,
d n n , , , ,

3.2e

orm a strong inductive conclusion for each of the following number sequences

1. , , , , 2. , , , , 3. , , , ,
36

3 ϫ
6 ϫ 3
ϫ 4

ϫ … 4. , , , ,
…
……
83
3

Chapter 3 Logical Reasoning Learning
Standard
How do you solve problems involving logical reasoning?
Solve problems involving
Example 23 logical reasoning.

The diagram on the right shows the growth of a cell which Cell A
begins with cell On the first day, two new cells are produced
CHAPTER 3 Every cell will produce two other cells on subsequent days
The number of cells growth is P(t t, where t is the number
of days.
a ow many new cells will be produced on the th day
b On which day will the number of new cells become

Solution:
a

Understanding the problem Planning a strategy
ubstitute t with into P(t t
i a e a conclusion by deduction
ii Calculate the number of new cells

on the th day
iii t
iv Calculate P

Conclusion Implementing the strategy

P(t P 8 Checking Answer
new cells will be produced on
, , , , , , ,
the th day

b

Understanding the problem Planning a strategy
olve t
i Calculate on which day the
number of new cells is Implementing the strategy
t
ii Calculate the value of t when t
P(t

Conclusion on

t
The number of new cells is
the th day

84

Chapter 3 Logical Reasoning

24
The table below shows the number of toy cars produced by the TO actory in a certain morning

Time Number of toy cars

a m

a m

a m CHAPTER 3

a m

a Construct a general formula for the number of toy cars produced by the TO actory based
on the above table

b The TO actory operates from o cloc in the morning until o cloc at night everyday
for days in a wee

i ow many toy cars can be produced by the TO actory in a day
ii The TO actory receives an order of toy cars This order needs to be completed

in a wee ill the TO actory be able to deliver the toy cars on time f it is not,
suggest a solution so that the TO actory can deliver this order

Solution:
a

Understanding the problem Planning a strategy

i a e a conclusion by induction Observe the pattern forms by the number of
ii Construct a general formula for toy cars produced

the number of toy cars produced

Conclusion Implementing the strategy

The general formula for the number
of toy cars produced by TO actory
is n n , , , ,






ence, the number of toy cars produced

can be formulated as

n n , , , ,

Saiz sebenar

85

Chapter 3 Logical Reasoning Planning a strategy

b i sing the general formula constructed in
part a , substitute n with
Understanding the problem
• The TO actory operates hours

a day
• n

CHAPTER 3 Conclusion toy cars Implementing the strategy

The TO actory produces
in a day

b ii Planning a strategy and

Understanding the problem • ultiply by
• Compare the results with
• The TO actory needs to produce
at least toy cars in days ma e a conclusion

• The TO actory produces
toy cars in a day

Conclusion Implementing the strategy
ϫ Ͻ
Therefore, the TO actory is not
able to deliver the toy cars on time

uggestion TO actory can extend
the operation hours to hours a day
so that on each day they can produce

toy cars

Checking Answer

ϫ
(Ͼ

Saiz sebenar

86

Chapter 3 Logical Reasoning

3.2f
1. The par ing charges at the Cahaya otel are calculated based on the rates below

Time Charge

The first hour or part of it R

Each hour thereafter until the th hour R

Each hour thereafter R

amuddin par ed his car from hours to attend a course at Cahaya otel fter the course, CHAPTER 3
amuddin too his car at hours Calculate by deduction the total charges that amuddin

has to pay prior to exiting from the car par

2. The number of residents in Taman embira follows the formula of g(t t t ,
where t is the number of years

t is given that the number of residents in Taman embira on anuary was

a a e a deductive conclusion about the total number of residents in Taman embira on
ecember

b n which year will the number of residents in Taman embira reach

3. The number of new born babies in a certain country in the year was The number
of new born babies in that country from the years to form a number pattern
as follows

Year New born babies
2014 536 100
2015 521 100
2016 506 100
2017 491 100

a Construct a formula based on the number pattern of new born babies

b f the number of new born babies in that country follows the above number pattern for the
next years, estimate the number of babies born in the year

87

Chapter 3 Logical Reasoning

4. The following diagram shows three right angled triangles
a Complete the table below

q

z p cb
x r
CHAPTER 3 a
y sin
cos sin
sin cos
cos

b Observe and state the relationship between the angles and values of sine and cosine functions
for each pair of angles above a e a conclusion by induction about the relationship
between sin θ and cos θ

c iven sin and based on the inductive conclusion from part b above, state the
value of cos

1. etermine whether each sentence below is a statement or not a statement ustify your answer
a cuboid has faces
b olve the equation x3 x x
c Each cylinder has two curved faces
d on t forget to bring wor boo tomorrow
e x
f a b a b a b
g ow, this flower is beautiful
h The members of P R are government officials
i Ͼ

2. etermine whether the following statements are true or false
f it is false, give one counter example
a x y x xy y
b ll integers have positive values
c fraction is smaller than one
d ll diagonals are perpendicular bisectors

88

Chapter 3 Logical Reasoning

3. etermine whether the following compound statements are true or false
a and ϫ
b and is a factor of
c , ʚ , , ∪ , or n(φ
d ϫ or ϫ ϫ

4. rite a true statement by using the quantifier all or some for the ob ects and characteristics
given below

Object Characteristic CHAPTER 3
a exagon as vertices

b Circle as a radius of cm
c Triangle as three axes of symmetry

5. a etermine the antecedents and consequents of the following statements

i f p Ͻ q, then q – p Ͼ
ii f the perimeter of rectangle A is x y , then the area of rectangle A is xy

b a e an appropriate implication based on each of the following pairs of implications

i f x is a multiple of , then x is a multiple of
f x is a multiple of , then x is a multiple of
ii f is a factor of , then is a factor of
f is a factor of , then is a factor of

c Construct two appropriate implications for each of the following implications

i of is if and only if ϫ
ii M is divisible by if and only if M is divisible by and

6. rite the requested statement in brac ets for each of the following and determine the truth
value of the written statement ustify your answer if it is false

a f and are two complementary angles, then Converse
b f w Ͼ , then w Ͼ Contrapositive
c f p Ͼ , then p Ͼ nverse
d The sum of exterior angles of a polygon is egation

7. Complete the following arguments to form valid and sound deductive arguments Saiz sebenar

a Premise ll factors of are factors of 89
Premise is a factor of
Conclusion
b Premise f x , then x
Premise
Conclusion x
c Premise
Premise sin cos β ≠
Conclusion α ≠ β

Chapter 3 Logical Reasoning

d Premise f p is divisible by , then p is a multiple of
Premise is divisible by
Conclusion
e Premise f m Ͻ , then m Ͼ
Premise
Conclusion m у
f Premise ll quadratic functions have a turning point
Premise
Conclusion The function g(x has a turning point

CHAPTER 3 8. a iven the surface area of a cone r(r s , ma e a conclusion by deduction for the surface
area of the five similar cones such that r cm and s cm

b iven the equation of a straight line is y mx c, ma e a conclusion by deduction for the
equation of the straight line PQ such that m and c

9. a e a conclusion by induction for the following number sequences

a , , , , b , , , ,





3
… ……
…

c , , , , d , , , ,













10. etermine whether the following arguments are inductive arguments or deductive arguments

a

ll the pupils from Class ahagia perform
on Teachers ay ayanthi is a pupil from
Class ahagia Then, ayanthi performs

on Teachers ay

b The sum of and is an even number
The sum of and is an even number
The sum of and is an even number
The sum of and is an even number
n conclusion, the sum of two odd numbers is an even number

90

Chapter 3 Logical Reasoning CHAPTER 3

11. The diagram below shows the number of cylinders of equal si e arranged in the boxes according
to a number pattern , , , ,

ox ox ox ox
a Construct a conclusion by induction for the pattern of the number of cylinders above
b f the radius and height of each cylinder are cm and cm respectively, calculate the

total volume of cylinders in the box
12. The diagram on the right shows the first four semicircles arranged according to a certain pattern

The radius of the largest semicircle is cm
a Calculate and list the perimeters of the four

semicircles, in terms of π
b ased on the answer from a , show that the

generalisation for the perimeters of the four
semicircles is n(π n , , , , ,
c C alculate the perimeter, in cm, of the th
semicircle
PROJ ECT

The rise in sea level is a critical issue for the whole world nowadays The rise in sea level
is closely related to the changes in temperature on the earth iven the equilibrium of sea
level and temperatures are connected by the formula, that is

L T T T

L is the change in sea level and T is the changes in temperature

Prepare a folio about the rise in sea level and changes in temperature for the last
five years our folio must contain
1. ront page
2. Content page
a n introduction to the issue of the rise in sea level
b conclusion by deduction about the changes in sea level for the last five years
c Reasons causing the rise in sea level
d mpact of human activity on the rise in sea level
e teps to control the rise in sea level
3. Conclusion

91

Chapter 3 Logical Reasoning

CONCEPT MAP
Logical Reasoning

tatements rguments
True or alse

CHAPTER 3 egation nductive rgument eductive rgument
o or ot trong and Cogent alid and ound

Example: a e general conclusion a e specific conclusion
is a multiple of based on specific premises based on general premises
egation is not a multiple
Example: Premise ll A is B
of Premise Premise C is A
Premise Conclusion C is B
Compound tatement Premise Example:
or or and Premise Premise ll multiples of
Conclusion are multiples of
Example: n n , , , , Premise is a multiple of
p is a multiple of Conclusion is a multiple of
q is a multiple of
• is a multiple of or Premise f p, then q
Premise p is true
a multiple of Conclusion q is true
• is a multiple of and
Example:
a multiple of Premise f x is a multiple of ,
then x is a multiple of
mplication Premise is a multiple of
f p, then q Conclusion is a multiple of
p if and only if q
Premise f p, then q
Example: Premise ot q is true
p is a multiple of Conclusion ot p is true
q is a multiple of
• f is a multiple of , Example:
Premise f x is a multiple of ,
then is a multiple of then x is a multiple of
• is a multiple of if and only Premise is not a multiple of
Conclusion is not a multiple of
if is a multiple of

Converse f q, then p

nverse f p, then q

Contrapositive f q, then p

Example:

f is a multiple of , then is a multiple of

Converse : f is a multiple of , then is

a multiple of

Inverse : f is not a multiple of , then

is not a multiple of

Contrapositive : f is not a multiple of , then

is not a multiple of

92

2. (a) True (b) False (c) False (d) False
(e) True (f) True (g) False (h) True

1. (a) 2405, 2415, 2425 Self Practice 3.1d
(b) 1102, 1112, 10002
(c) 317, 327, 337 1. (a) If x = 3, then x4 = 81.

2. 32 (b) If ax3 + bx2 + cx + d = 0 is a cubic equation, then

3. (a) 7168 a ≠
(b) 111101112
(b) 14315 (c) If n – 5 Ͼ 2n, then n Ͻ –5.
4. (a) 111100012 (d) 3618 m
(c) 4637 (b) 4427 (d) If n Ͼ 1, then m2 Ͼ n2.
(b) True
5. (a) 101012 (c) 569 2. (a) Antecedent: x is an even number.
6. (a) True (c) False
Consequent: x2 is an even number.

7. 269 (b) Antecedent: set K = φ.

8. 39 Consequent: n(K) = 0.

9. y = 105 (c) Antecedent: x is a whole number.

10. (a) 658, 1101102 Consequent: 2x is an even number.
(b) 1768, 10035
(d) Antecedent: A straight line AB is a tangent to
11. 1325
12. 558 a circle P.
13. 427
Consequent: A straight line AB touches the circle

P at one point only.

3. (a) k is a perfect square if and only if k is a whole

CHAPTER 3 Logical Reasoning number.

Self Practice 3.1a (b) P ʝ Q = P if and only if P ʚ Q.

(c) pq = 1 if and only if p = q–1 and q = p–1.

1. (a) Not a statement. Because the truth value cannot (d) k2 = 4 if and only if (k + 2)(k – 2) = 0.

be determined. 4. (a) If PQR is a regular polygon, then PQ = QR = PR.

(b) A statement. Because it is true. If PQ = QR = PR, then PQR is a regular polygon.

(c) Not a statement. Because the truth value cannot (b) If m is an improper fraction, then m Ͼ n.
n
be determined. m
If m Ͼ n, then n is an improper fraction.
(d) A statement. Because it is true.
(c) If 9 is the y-intercept of a straight line y = mx + c,
(e) Not a statement. Because the truth value cannot
then c = 9.
be determined.
If c = 9, then 9 is the y-intercept of a straight line
2. (a) 40 Ͼ 23 + 9
y = mx + c.
(b) {3} ʚ {3, 6, 9}
(d) If f (x) = ax2 + bx + c has a maximum point, then
(c) 1 10 = 5
4 3 6 a Ͻ 0.

(d) x2 + 3 ഛ (x + 3)2 If a Ͻ 0, then f (x) = ax2 + bx + c has a maximum

(e) 3 27 + 9 = 12 point.

3. (a) False (b) False (c) False Self Practice 3.1e

(d) True (e) True

1. (a) Converse: If x Ͼ –1, then x + 3 Ͼ 2.

Self Practice 3.1b Inverse: If x + 3 ഛ 2, then x ഛ –1.

1. 819 is not a multiple of 9. False Contrapositive: If x ഛ –1, then x + 3 ഛ 2.
2. A kite does not have two axes of symmetry. True
3. A cone does not have one curved surface. False (b) Converse: If k = 3 or k = –4, then
4. Two parallel lines do not have the
False (k – 3)(k + 4) = 0.
same gradient.
5. Not all quadratic equations have two True Inverse: If (k – 3)(k ≠ , then k ≠ or

equal roots. k ≠

Contrapositive: If k ≠ or k ≠ , then

(k – 3)(k ≠

(c) Converse: If AB is parallel to CD, then

Self Practice 3.1c ABCD is a parallelogram.

1. (a) 2 or 3 is a prime factor of the number 6. Inverse: If ABCD is not a parallelogram,
(b) A cone has one vertex and one plane.
(c) A rhombus and a trapezium are parallelograms. then AB is not parallel to CD.

Contrapositive: If AB is not parallel to CD, then

ABCD is not a parallelogram.

295

2. (a) Implication: If 2 and 5 are the factors True (d) If a triangle ABC does not have a right angle at C,
True then c2 a2 + b2. True
of 10, then 2 × 5 is 10. True
Converse: If 2 × 5 is 10, then 2 and 5 True (e) If w ജ 5, then w ജ 7. False. When w = 6,
are the factors of 10. 6 Ͼ 5 but 6 Ͻ 7.

Inverse: If 2 and 5 are not the factors Self Practice 3.2a 6. Inductive argument
7. Inductive argument
of 10, then 2 × 5 is not 10. 1. Deductive argument 8. Deductive argument
Contrapositive: If 2 × 5 is not 10, then 2. Inductive argument 9. Deductive argument
2 and 5 are not the factors of 10. 3. Inductive argument 10. Inductive argument
4. Deductive argument
(b) Implication: If 4 is a root of x2 – 16 = 0, False 5. Deductive argument
then 4 is not a root of True
(x + 4)(x – 4) = 0. True Self Practice 3.2b
Converse: If 4 is not a root of False
(x + 4) (x – 4) = 0, then 4 is a root of 1. Valid but not sound because premise 1 and conclusion
x2 – 16 = 0. are not true.
Inverse: If 4 is not a root of x2 – 16 = 0,
then 4 is a root of (x + 4)(x – 4) = 0. 2. Valid and sound
Contrapositive: If 4 is a root of 3. Valid and sound
(x + 4)(x – 4) = 0, then 4 is not a root 4. Valid but not sound because premise 1 is not true.
of x2 – 16 = 0. 5. Not valid because the conclusion is not formed based

(c) Implication: If a rectangle has four True on the given premises. Hence, the argument is not sound.
axes of symmetry, then the rectangle False 6. Valid and sound
has four sides. False 7. Not valid because the conclusion is not formed based
Converse: If a rectangle has four True
sides, then the rectangle has four axes on the given premises. Hence, the argument is not sound.
of symmetry. 8. Valid and sound
Inverse: If a rectangle does not have 9. Not valid because the conclusion is not formed based
four axes of symmetry, then the
rectangle does not have four sides. on the given premises. Hence, the argument is not sound.
Contrapositive: If a rectangle does 10. Valid and sound
not have four sides, then the rectangle
does not have four axes of symmetry. Self Practice 3.2c

(d) Implication: If 55 + 55 = 4 × 5, then True 1. (a) Preevena uses digital textbook.
(b) Kai Meng gets a cash prize of RM200.
666 + 666 = 6 × 6 True (c) Quadrilateral PQRS is not a regular polygon.
(d) ABC has one axis of symmetry.
Converse: If 666 + 666 = 6 × 6, (e) m : n = 2 : 3
(f) m + 3 Ͼ 2m – 9
then 55 + 55 = 4 × 5.
2. (a) Straight line AB has zero gradient.
Inverse: If 55 + 55 4 × 5, then True (b) All multiples of 9 are divisible by 3.
(c) Polygon P is a nonagon.
666 + 666 6 × 6. (d) If x Ͼ 6, then x Ͼ 4.
Contrapositive: If 666 + 666 6 × 6, True (e) The room temperature is not lower than 19°C.
(f) If 3x – 8 =16, then x = 8.
then 55 + 55 4 × 5.

Self Practice 3.1f Self Practice 3.2d

1. (a) False. A rectangle does not have four sides of 1. This argument is weak and not cogent because the
equal length. conclusion is probably false.

(b) True 2. This argument is strong and cogent.
(c) True 3. This argument is weak and not cogent because the
(d) False. 36 is not divisible by 14.
2. (a) 1008 – 778 18. False because 1008 – 778 = 18. conclusion is probably false.
(b) A cuboid does not have four uniform cross 4. This argument is strong and cogent.
5. This argument is strong but not cogent because
sections. True
(c) If y = 2x and y = 2x – 1 have the same gradient, premise 3 is false.
6. This argument is weak and not cogent because the
then y = 2x is parallel to y = 2x – 1. True
conclusion is probably false.

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Self Practice 3.2e (ii) If M is divisible by 20, then M is divisible by
2 and 10.
1. (3n)–1; n = 1, 2, 3, 4, … If M divisible by 2 and 10, then M is divisible
2. 5n; n = 1, 2, 3, 4, ... by 20.
3. 2(n)3 + n; n = 0, 1, 2, 3, ...
4. 20 – 4n; n = 0, 1, 2, 3, ... 6. a f , then and are two complementary
angles. True
Self Practice 3.2f
(b) If w ഛ 30, then w ഛ 20. False because 28 Ͻ 30
1. RM43 but 28 Ͼ 20.

2. (a) 32 500 residents (b) 14th year (c) If p ഛ 0, then p2 ഛ 0. False because –2 Ͻ 0 but
(–2)2 Ͼ 0.
3. (a) 536 100 – 15 000n (b) 431 100 babies a
y p ca (d) The sum of exterior angles of a poligon is not
4. (a) sin 60° = zy sin 40° = rp sin 20° = c 360°. False because the sum of exterior angles in
cos 30° = z cos 50° = r cos 70° = each polygon is 360°.

(b) sin = cos (90° – ) 7. (a) 2 is a factor of 8.
(b) x = 5
(c) 0.9848
c f , then sin2 cos2
1. (a) A statement because it is true. (d) 54 is a multiple of 18.
(e) m ഛ 0
(b) Not a statement because the truth value cannot be (f) The function g(x) is a quadratic function.

determined. 8. a The surface area of the five similar cones is
700 cm2.
(c) A statement because it is false.
(b) The equation of the straight line PQ is y = 3x + 5.
(d) Not a statement because the truth value cannot be 9. (a) n2 – 5 ; n = 1, 2, 3, 4, ...

determined. (b) 2n + 3 ; n = 0, 1, 2, 3, ...
(c) 4n + n2 ; n = 1, 2, 3, 4, ...
(e) Not a statement because the truth value cannot be (d) 3n + 2(n – 1)2 ; n = 1, 2, 3, 4, ..
10. (a) Deductive argument
determined. (b) Inductive argument
11. (a) The pattern of the number of cylinders is 2n +1;
(f) A statement because it is true.
n = 1, 2, 3, 4, ...
(g) Not a statement because the truth value cannot be (b) 104 720 cm3
12. (a) 32( + 2), 16( + 2), 8( + 2), 4( + 2)
determined.

(h) A statement because it is true.

(i) A statement because it is false. (c) 1 ( + 2) cm
4
2. (a) True

(b) False. –3 is an integer with negative value.

(c) False. 3 is a fraction larger than one. CHAPTER 4 Operations on Sets
2

(d) False. The diagonals of a kite are not Self Practice 4.1a

a perpendicular bisector. 1. (a) M = {1, 3, 5, 7, 9}

3. (a) False (b) N = {3, 6, 9}

(b) True

(c) True (c) M ʝ N = {3, 9}

(d) False 2. (a) J ʝ K = {4, 6, 9} (b) J ʝ L = {3, 9}

4. (a) All hexagons have 6 vertices. (c) K ʝ L = {9} (d) J ʝ K ʝ L = {9}

(b) Some circles have a radius of 18 cm. 3. (a), (b)

(c) Some triangles have three axes of symmetry. P • • •

5. (a) (i) Antecedent: p Ͻ q • • Q ••

Consequent: q – p Ͼ 0 •
•
(ii) Antecedent: The perimeter of rectangle A is • • ••

2(x + y). • • ••

Consequent: The area of rectangle A is xy. R • ••

(b) (i) x is a multiple of 10 if and only if x is

a multiple of 5. 4. (a) A ʝ B = {I}, n(A ʝ B) = 1
(b) A ʝ C = φ, n(A ʝ C) = 0
(ii) 6 is a factor of 12 if and only if 6 is a factor of (c) B ʝ C = φ, n(B ʝ C) = 0
(d) A ʝ B ʝ C = φ, n(A ʝ B ʝ C) = 0
24.

(c) (i) If 20% of 30 is 6, then 0.2 × 30 = 6.
If 0.2 × 30 = 6, then 20% of 30 is 6.

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