CHAPTER 3: LOGICAL REASONING

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


Sentence
Statement
Not a statement
Please send the workbook
/
Kuala Lumpur Tower is the tallest tower in Malaysia
/
How do you come to school?
/
x + 3 =5
/
-6 < -8
/
23 is a prime number.
Please stand up.
800 ml = 0.08 l
x +5
2k = 8
How old are you?


Aim : To determine the truth values of the given statements.


Mathematical Sentence
Truth Value
28 + 12 = 40
True
32 + 42 =72
False
(2+3) (2-3) =22 – 32
True
√729 = 813/2
False
(x – y)2 = x2 – 2xy + y2
True
{a, b } has 4 subsets
True
5 is a factor of 400
True
The lowest common multiple of the numbers 4 and 18 is 36.
True


We use the word “no” or “not” to negate a statement. The negation of statement p is written as ~p


STATEMENT ( p )
NEGATION (~p )
12 is a multiple of 5
12 is not a multiple of 5
41 is a prime number
41 is not a prime number
All multiples of 5 are multiples of 10
Not all multiples of are multiples of 10
All decimal numbers are less than 1
Not all decimal numbers are less than 1.
34 ≠ 44
34 = 44


A COMPOUND STATEMENT IS A COMBINATION OF TWO OR MORE STATEMENTS BY USING THE WORD “AND” OR “OR” .
P AND Q P OR Q


p : A pentagon has two diagonals q : A heptagon has four diagonals
i. A pentagon has two diagonals and a heptagon has four diagonals.
ii. A pentagon has two diagonals or a heptagon has four diagonals.


5 and 7 are odd numbers p : 5 is an odd number.
q : 7 is an odd number
22 = 4 or 23 = 8
p : 22 = 4 q : 23 = 8


P
Q
P AND Q
P OR Q
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE


2 and -5 are greater than 4
p : 2 is greater than 4 (FALSE)
q : - 5 is greater than 4 (FALSE) P AND Q : FALSE


32 + 42 = 52 AND 62 + 82 = 102 p : 32 + 42 = 52 (TRUE)
q : 62 + 82 = 102 (TRUE) P AND Q :TRUE


- 3 > - 4 or – 4 > - 3
p : - 3 > - 4 (TRUE) q : – 4 > - 3 (FALSE)
P OR Q : TRUE


24 is a multiple of 4 or 7
p : 24 is a multiple of 4 (TRUE) q : 24 is a multiple of 7 (FALSE)
P OR Q : TRUE


1. IMPLICATION “IF P , THEN Q”
2. IMPLICATION “P IF AND ONLY IF Q”


A STATEMENT “ IF P, THEN Q” IS KNOWN AS IMPLICATION.
P IS DENOTED AS ANTECEDENT Q IS DENOTED AS THE
CONSEQUENT


Antecedent (p) : k is divisible by 5 Consequent (q) : k is multiple of 5
If k is divisible by 5, then k is multiple of 5.


Antecedent (p) : ab = 0 Consequent (q) : a = 0 or b = 0
If ab = 0, then a = 0 or b = 0


Identify the antecedent and consequent of each the following implication.
a. If θis an obtuse angle, then 90° < θ< 180°
b. If x + y > 0, then y > - x
Solution :
a) Antecedent : θis an obtuse angle
Consequent : 90° < θ< 180°
b) Antecedent : x + y > Consequent : y > - x


An implication “ p if and only if q” consist of the following two implications.
• If p, then q •If q, then p


Write two implications from each of the following compound statement.
5x = 5 if and only if x = 1
Implication 1: If 5x = 5, then x = 1 Implication 2 : If x = 1, then 5x = 5


Write two implications from each of the following compound statement.
y = 9 if and only if √y = 3
Implication 1: If y = 9 , then √y = 3 Implication 2 : If √y = 3, then y = 9


Based on the given antecedent and consequent , construct a mathematical statement in the form of :
i)If p, then q ii)p if and only if q
1) Antecedent : x is an even number Consequent : x is divisible by 2
2) Antecedent : m2 = 16 Consequent : m = ± 4


IN GENERAL
Statement Converse Inverse Contrapositive
: If p , then q
: If q, then p
: If ~p , then ~q : If ~q, then ~p


GENERA
L TYPE OF
STATEMENT
SPECIFIC


The area of triangle ABC is 8 cm2
• Specific
All multiples of 2 end with even digits
• General


All numbers in base 5 consist digits which smaller than 5
GENERAL
15 is divisible by 3
SPECIFIC


All prisms have uniform cross sections
The volume of a sphere is 4/3πr3, where r is the radius of the sphere
GENERAL
GENERAL


ARGUMENTS
DEDUCTIVE INDUCTIVE


Process of making a
specific conclusion based on general premises
Example:
The area of a circle is πr2. Circle A has a radius of 7 cm. In conclusion, the area of circle A is 154 cm2.
DEDUCTIVE ARGUMENT
Premise 1 :
The area of a circle is πr2.
(General Premise)


 Process of making a general conclusion based on specific premises
Example:
The sum of the digits in 18 is divisible by 9.
The sum of the digits in 27 divisible by 9.
Thus , the sum of the digits in multiples of 9 is divisible by 9.
Premise 1 :
The sum of the digits in 18 is divisible by 9.
(Specific Premise)
Thus , the sum of the digits in multiples of 9 is divisible by 9. (General Conclusion)
INDUCTIVE ARGUMENT




SELF PRACTICE 3.2A PAGE 74


FORM 1
FORM II
FORM III
Premise 1
All A are B
If p, then q
If p ,then q
Premise 2
C is A
p is true
Not q is true
Conclusion
C is B
q is true
Not p is true
ALL deductive argument is said to be valid if all the premises and the conclusion are true


FORM 1
Premise 1
All A are B
Premise 2
C is A
Conclusion
C is B
PREMISE 1 PREMISE II CONCLUSION
AB
: All actresses are good at dancing
CA
: Jasmine is an actress
CB
: Jasmine is good at dancing
Above argument is a valid argument.
But the argument is unsound because Premise 1 is false.


PREMISE II : Jasmine is good at dancing
CONCLUSION : Jasmine is an actress
A
B
PREMISE 1 : All actresses are good at dancing CB
CA
WHY IS THE ABOVE ARGUMENT NOT VALID?
Above argument is not valid argument.
It does not comply with the valid of FORM I (deductive argument)


FORM II
Premise 1
If p, then q
Premise 2
p is true
Conclusion
q is true
PREMISE 2 : Polygon Q is a pentagon.
P
Q
PREMISE 1 : If a polygon is a pentagon, then the sum of the interior angles of the polygon is 540°
P is true
CONCLUSION: The sum of the interior angles of polygon Q is 540°
Q is true


FORM III
Premise 1
If p ,then q
Premise 2
Not q is true
Conclusion
Not p is true
P
Q
PREMISE 1: If y + 4 = 10, then y = 6
PREMISE 1 :y ≠6 CONCLUSION : y + 4 ≠ 10
Q is not true
P is not true


Complete of the following arguments:
PREMISE 1 : All multiples of 10 are divisible by 5 PREMISE 2: M is a multiple of 10.
CONCLUSION : M is divisible by 5


PREMISE 1 : All trapezium have two parallel sides.
PREMISE 2: ABCD is a trapezium CONCLUSION : ABCD has two parallel sides.


PREMISE 1: If x = 6, then x + 7 = 13 PREMISE 2 :x = 6
CONCLUSION : x + 7 = 13


PREMISE 1 : If m is a multiple of 2, then m is an even number.
PREMISE 2 : 24 is a multiple of 2.
CONCLUSION :
24 is an even number


PREMISE 1 : If 4x = -8, then x = -2
PREMISE 2 : CONCLUSION : 4x ≠ - 8
x ≠ -2


SELF PRACTICE 3.2 B : page 77 QUESTION 1 ,3, 5, 7, 9
SELF PRACTICE 3.2C : page 79 QUESTION 1 (a, b, c) QUESTION 2 (a, b, c)


HOW DO YOU DETERMINE AND JUSTIFY THE STRENGTH OF AN INDUCTIVE AND HENCE
DETERMINE STRONG COGENT?
WHETHER THE ARGUMENT IS


INDUCTIVE ARGUMENT
STRONG WEAK
ALL PREMISES ARE FALSE
NOT NOT
TRUE CONCLUSION
FALSE CONCLUSION
ALL PREMISES ARE TRUE
COGENT
COGENT
COGENT


Premise 1 : The chairs in the living room are red. Premise 2 : The chairs in the dining room are red. Premise 3 : The chairs in the study room are red. Premise 4 : The chairs in the bedroom are red. Conclusion : All chairs in the house are red.
This argument is weak and not cogent because although the premises are true, the conclusion is probably false.


Premise 1 : 27 is a multiple of 3.
Premise 2 : 81 is multiple of 3.
Conclusion : All multiples of 9 are multiples of 3.
This argument is strong and cogent because all the premises and conclusion are true.


EXERCISE 3.2 D (PAGE 82)