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Mathematical Reasoning

4.1 Statements (Sample Questions)

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Example 1:
Determine whether the following sentences are statements or not. Give a reason for your answer.
(a) 3 + 3 = 8
(b) 9 – 4 = 5
(c) A pentagon has 5 sides.
(d) 4 is a prime number.
(e) Is 40 divisible by 3?
(f) Find the perimeter of a square with each side of 4 cm.
(g) Help!

Answer:
(a) Statement; it is a false statement.
(b) Statement; it is a true statement.
(c) Statement; it is a true statement.
(d) Statement; it is a false statement.
(e) Not a statement; it is a question.
(f) Not a statement; it is an instruction.
(g) Not a statement; it is an exclamation.





Deduction and Induction

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4.6 Deduction and Induction

(A) Reasoning by Deduction and Induction
1. Reasoning by deduction is a process of making a conclusion for a specific case based on a given general statement.

2. Reasoning by induction is a process of making a generalization based on specific cases.

Maths Tip
1. General statement  →  Special conclusion  → Deduction
2. Specific cases  →  General conclusion  →  Induction
Example: 
Determine whether the following conclusion is made based on a deductive reasoning or inductive reasoning.

(a) Area of triangle = ½ × Base × Height
(i) 

Area of ∆ ABC
= ½ × 7cm × 5cm
= 17.5 cm2 
  
(ii)

Area of ∆ DEF
= ½ × 7cm × 4cm
= 14 cm2
  
(b)
1 = 7 (1)2 – 6
22 = 7 (2)2 – 6
57 = 7 (3)2 – 6
106 = 7 (4)2 – 6
 7n2 – 6, n = 1, 2, 3, 4…
 
Solution: 
(a)
The specific conclusion is made based on a general statement ~ Area of triangle = ½ × Base × Height. Therefore, the conclusion is made based on deductive reasoning.

(b) 
The general conclusion 7n2 – 6, n = 1, 2, 3, 4… is made based on specific cases. Therefore, the conclusion is based on inductive reasoning.

4.4 Implication

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(A) Antecedent and Consequent of an Implication

  1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.
  2. p is called the antecedent.
    q is called the consequent.


Example:

Identify the antecedent and consequent of the following implications.

(a) If m = 2, then 2m2 + m = 10

(b) I f P ∪ Q = P , t h e n Q ⊂ P

Solution:

  1. Antecedent: m = 2
    Consequent: 2m2 + m = 10
  2. Antecedent : P ∪ Q = P Consequent : Q ⊂ P

(B) Implications of the Form ‘p if and only if q’

  1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.
  2. Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:
    Implication 1: If p, then q.
    Implication 2: If q, then p.


Example 1:

Given that p: x + 1 = 8
q: x = 7
Construct a mathematical statement in the form of implication

  1. If p, then q.
  2. p if and only if q.

Solution:

  1. If x + 1 = 8, then x = 7.
  2. x + 1 = 8 if and only if x = 7.

Example 2:

Write down two implications based on the following sentence:
x3 = 64 if and only if x = 4.

Solution:

If x3 = 64, then x = 4.
If x = 4, then x3 = 64.

(C) Converse of an Implication

  1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.

Example:

State the converse of each of the following implications.

  1. If x2 + x – 2 = 0, then (x – 1)(x + 2) = 0.
  2. If x = 7, then x + 2 = 9.

Solution:

  1. If (x – 1)(x + 2) = 0, then x2 + x – 2.
  2. If x + 2 = 9, then x = 7.

 

 

4.6 SPM Practice (Long Questions)

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Question 7:
(a)(i) State whether the following compound statement is true or false.

3 + 3 = 9 or   3 × 3 = 9
(a)(ii) Determine whether the following converse is true or false.

If x > 3, then x > 7
(b) Write down Premise 2 to complete the following argument:
Premise 1: If y = mx + 5 is a linear equation, then m is a gradient of the straight line.
Premise 2: _____________________
Conclusion: 2 is the gradient of the straight line.

(c) Angle subtended at the centre of a regular polygon with n sides is 360 o n .
Make one conclusion by deduction for the angle subtended at the centre of a regular polygon with 5 sides.

Solution:
(a)(i) True

(a)(ii) The converse is true

(b) Premise 2: y = 2x + 5 is a linear equation

(c)
Angle subtended at the centre of a regular pentagon = 360 o n = 360 o 5 = 72 o


Question 8:
(a) State whether the following sentence is a statement or not a statement.

x + 7 = 9
(b) Complete the following compound statement by writing the word ‘or’ or ‘and’ to form a true statement.
23 = 6 …… 5 × 0 = 0

(c) Write down Premise 2 to complete the following argument:
Premise 1: All isosceles triangles have two equal sides.
Premise 2: _____________________
Conclusion: ABC has two equal sides.

(d) Make a general conclusion by induction for the sequence of numbers 1, 7, 17, 31, … which follows the following pattern.
1 = (2 × 1) – 1
7 = (2 × 4) – 1
17 = (2 × 9) – 1
31 = (2 × 16) – 1

Solution:
(a) Not a statement

(b) 23 = 6 …or… 5 × 0 = 0

(c) ABC is an isosceles triangle.

(d)
1 = (2 × 1) – 1 =  (2 × 12) – 1
7 = (2 × 4) – 1 =  (2 × 22) – 1
17 = (2 × 9) – 1 =  (2 × 32) – 1
31 = (2 × 16) – 1 =  (2 × 42) – 1
  =  2n2 – 1, n = 1, 2, 3, …

4.5 Arguments

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(A) Premises and Conclusions

  1. An argument is a process of making conclusion based on several given statements.
  2. The statements given are known as premises.
  3. An argument consists of premises and a conclusion.

Example 1:

Identify the premises and conclusion of the following argument.

  1. A pentagon has 5 sides. ABCDE is a pentagon. Therefore, ABCDE has 5 sides.

Solution:

Premise 1: A pentagon has 5 sides.
Premise 2: ABCDE is a pentagon.
Conclusion: ABCDE has 5 sides.

(B) Forms of Arguments

  1. Based on two given premises, a conclusion can be made for three different forms of arguments.

Argument Form I

Premise 1: All A are B.
Premise 2: C is A.
Conclusion: C is B.

Example 2:

Make a conclusion based on the two premises given below.

Premise 1: All multiples of 5 are divisible by 5.
Premise 2: 45 is a multiple of 5.
Conclusion:  _______________

Solution:

Conclusion: 45 is divisible by 5.

Argument Form II

Premise 1: If p, then q.
Premise 2: p is true.
Conclusion: q is true.

Example 3:

Make a conclusion based on the two premises given below.
Premise 1: If a number is a factor of 18, then the number is a factor of 54.
Premise 2: 3 is a factor of 18.
Conclusion:  _______________

Solution:

Conclusion: 3 is a factor of 54.

Argument Form III

Premise 1: If p, then q.
Premise 2: Not q is true.
Conclusion: Not p is true.

Example 4:

Make a conclusion based on the two premises given below.
Premise 1: If P is a subset of Q, then P ∩ Q = P .
Premise 2: P ∩ Q ≠ P
Conclusion:  _______________

Solution:

Conclusion: P is not the subset of Q.

 

 

4.6 SPM Practice (Long Questions)

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Question 5:
(a) State if each of the following statements is true or false.
 (i)    23= 8 or ⅓ = 1.33.
 (ii)  – 6 > – 8 and 6 > 8.
 
(b) Write down two implications based on the following statement:

x a + y b =1 if and only if bx+ay=ab.

(c) It is given that the interior angle of a regular polygon of n sides 
  is   ( 1 − 2 n ) × 180 ∘ .
  Make one conclusion by deduction on the size of the interior angle of a regular hexagon.

Solution:
(a)(i) True

(a)(ii) False

(b) 
Implication 1:   If   x a + y b =1, then bx+ay=ab. _ Implication 2:   If  bx+ay=ab, then  x a + y b =1. _

(c)
Size of an interior angle of a regular hexagon = ( 1 − 2 6 ) × 180 ∘ = 2 3 × 180 ∘ = 120 ∘


Question 6:
(a) Complete the following mathematical sentence by writing the symbol > or <.
  (i) 53____ 20 is a false statement.
  (ii) – 3 ____ – 10 is a true statement.

(b) Complete the conclusion in the following argument:
 Premise 1 : If  n 1 2 = n , then  4 1 2 = 4 =2.   Premise 2 :  n 1 2 = n  Conclusion : _____________________
(c) Make a general conclusion by induction for the sequence of numbers 10, 35, 70, … which follows the following pattern.

10 = 5 (2)2 – 10 
35 = 5 (3)2 – 10
70 = 5 (4)2 – 10
…. = ………..

Solution:
(a)(i) 53   <    20 is a false statement.

(a)(ii)   – 3    >    – 10 is a true statement.

(b)
Conclusion : 4 1 2 = 4 = 2

(c) 5 (n + 1)2 – 10, where n = 1, 2, 3, …


4.4 Implications Short Notes

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4.4 Implications

(A) Antecedent and Consequent of an Implication
1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.

2. p is called the antecedent.
   q is called the consequent.

Example:
Identify the antecedent and consequent of the following implications.
(a) If m = 2, then 2m2 + m = 10
(b) If P∪Q=P, then Q⊂P

Solution:
(a) Antecedent: m = 2
Consequent:: 2m2 + m = 10

(b) Antecedent:P∪Q=P       Consequent:Q⊂P


(B) Implications of the Form ‘p if and only if q’
1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.
2. Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:
Implication 1: If p, then q.
Implication 2: If q, then p.

Example 1:
Given that p: x + 1 = 8
 q: x = 7
Construct a mathematical statement in the form of implication
(a) If p, then q.
(b) p if and only if q.

Solution:
(a) If x + 1 = 8, then x = 7.
(b) x + 1 = 8 if and only if x = 7. 

Example 2:
Write down two implications based on the following sentence:
x3 = 64 if and only if x = 4.

Solution:
If x3 = 64, then x = 4.
If x = 4, then x3 = 64.



(C) Converse of an Implication
1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.
 
Example:
State the converse of each of the following implications.
(a) If x2 + x – 2 = 0, then (x - 1)(x + 2) = 0.
(b) If x = 7, then x + 2 = 9.

Solution:
(a) If (x - 1)(x + 2) = 0, then x2 + x – 2.
(b) If x + 2 = 9, then x = 7.

4.6 SPM Practice (Long Questions)

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Question 3:
(a) Complete each of the following statements with the quantifier ‘all’ or ‘some’ so that it will become a true statement.
  (i) ___________ of the prime numbers are odd numbers.
 (ii) ___________ pentagons have five sides.

(b) Write down two implications based on the following statement:

A ∩ B = B if and only if A υ B = A.
(c) Complete the premise in the following argument:

Premise 1: If a number is a factor of 24, then it is a factor of 48.
Premise 2: 12 is a factor of 24.
Conclusion: _____________________

Solution:
 (a)(i)    Some  of the prime numbers are odd numbers.
 (a)(ii)  All pentagons have five sides.

(b) Implication 1: If A ∩ B = B, then A υ B = A.
  Implication 2: If A υ B = A, then A ∩ B = B.

(c) Conclusion: 12 is a factor of 48.


Question 4:
(a) Combine the following two statements to form one true statement.
  Statement 1: (– 3)² = 9
  Statement 2: –3 (3) = 19
(b) Complete the premise in the following argument:

Premise 1: _____________________
Premise 2: x is a multiple of 25.
Conclusion: x is a divisible of 5.
(c) Make a general conclusion by induction for the sequence of numbers 7, 14, 27, … which follows the following pattern.
7 = 3 (2)1 + 1
14 = 3 (2)2 + 2
27 = 3 (2)3 + 3
…. = ………..
Solution:
(a) (– 3)² = 9 or –3 (3) = 19.

(b) Premise 1: All multiples of 25 is divisible by 5.

(c) 3 (2)n + n, where n = 1, 2, 3, …

4.2 Quantifiers ‘All’ and ‘Some’ (Sample Questions)

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Example 1:
Complete each of the following statements using the quantifiers ‘all’ or ‘some’ to make the statement true.
(a) _____ rectangles are squares.
(b) _____ prime numbers are odd numbers.
(c) _____ triangles have equal sides.
(d) _____ even numbers are divisible by 2.

Answer:
(a) Some rectangles are squares.
(b) Some prime numbers are odd numbers.
(c) Some triangles have equal sides.
(d) All even numbers are divisible by 2.



Example 2:
Construct a true statement using the quantifier ‘all’ or ‘some’ for the given object and property.
(a) Object: multiples of 4
 Property: can be divided exactly by 5
(b) Object: regular hexagon
 Property: 6 equal sides
(c) Object: acute angles
 Property: less than 90o

Answer:
(a) Some multiples of 4 can be divided exactly by 5.
(b) All regular hexagons have 6 equal sides.
(c) All acute angles are less than 90o.

4.5 Arguments (Short Notes)

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4.5 Arguments

(A) Premises and Conclusions
1. An argument is a process of making conclusion based on several given statements.
2. The statements given are known as premises.
3. An argument consists of premises and a conclusion.

Example 1:
Identify the premises and conclusion of the following argument.
(a) A pentagon has 5 sides. ABCDE is a pentagon. Therefore, ABCDE has 5 sides.

Solution:
Premise 1: A pentagon has 5 sides.
Premise 2: ABCDE is a pentagon.
Conclusion: ABCDE has 5 sides.


(B) Forms of Arguments
1. Based on two given premises, a conclusion can be made for three different forms of arguments.

Argument Form I
Premise 1: All A are B.
Premise 2: C is A.
Conclusion: C is B.
Example 2:
Make a conclusion based on the two premises given below.
Premise 1: All multiples of 5 are divisible by 5.
Premise 2: 45 is a multiple of 5.
Conclusion: ______________

Solution:
Conclusion: 45 is divisible by 5.

Argument Form II
Premise 1: If p, then q.
Premise 2: p is true.
Conclusion: q is true.
Example 3:
Make a conclusion based on the two premises given below.
Premise 1: If a number is a factor of 18, then the number is a factor of 54.
Premise 2: 3 is a factor of 18.
Conclusion: ______________

Solution:
Conclusion: 3 is a factor of 54.

Argument Form III
Premise 1: If p, then q.
Premise 2: Not q is true.
Conclusion: Not p is true.

Example 4:
Make a conclusion based on the two premises given below.
Premise 1: If P is a subset of Q, then P  ∩ Q = P.
Premise 2: P  ∩ Q  ≠ P
Conclusion: ______________

Solution:
Conclusion: P is not the subset of Q.