Lecture 2 - Principle of Mathematical Induction and Well Ordering Principle | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths 
Lesson: Principle of Mathematical Induction and Well 
Ordering Principle 
Course Developer: Dr. Roopesh Tehri 
College: Acharya Narendra Dev College, (D.U.) 
  
Page 2


Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths 
Lesson: Principle of Mathematical Induction and Well 
Ordering Principle 
Course Developer: Dr. Roopesh Tehri 
College: Acharya Narendra Dev College, (D.U.) 
  
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
 
 
 
 
 
Table of Contents 
 Chapter : Principle of Mathematical Induction 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Principle of Mathematical Induction 
o 3.1. First Form of Principle of Mathematical Induction 
o 3.2. Second Form of Principle of Mathematical Induction 
? 4. Well Ordering Principle 
o 4.1. Equivalence of Principle of Mathematical Induction 
(PMI) and Well Ordering Principle (WOP) 
o 4.2. Euclidean Algorithm 
? 5. Division Algorithm 
? 6. Fundamental Theorem of Arithmetic 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
1. Learning Outcomes: 
Page 3


Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths 
Lesson: Principle of Mathematical Induction and Well 
Ordering Principle 
Course Developer: Dr. Roopesh Tehri 
College: Acharya Narendra Dev College, (D.U.) 
  
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
 
 
 
 
 
Table of Contents 
 Chapter : Principle of Mathematical Induction 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Principle of Mathematical Induction 
o 3.1. First Form of Principle of Mathematical Induction 
o 3.2. Second Form of Principle of Mathematical Induction 
? 4. Well Ordering Principle 
o 4.1. Equivalence of Principle of Mathematical Induction 
(PMI) and Well Ordering Principle (WOP) 
o 4.2. Euclidean Algorithm 
? 5. Division Algorithm 
? 6. Fundamental Theorem of Arithmetic 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
1. Learning Outcomes: 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
After studying the whole contents of this chapter, students will be able to 
understand:  
? The two forms of PMI with the help of suitable examples. 
? The statement of Well Ordering Principle and its equivalence with PMI. 
? The concept of basic definitions like divisibility, gcd , lcm ,Euclidean 
Algorithm etc  in integers with the help of suitable examples. 
? The concept of division Algorithm. 
? Fundamental Theorem of Arithmetic with its proof.  
 
2. Introduction: 
For future understanding of the abstract algebra, the properties of integers 
are very much handy and useful. In this lesson, we studied the basic 
properties of integers, starting with the two forms of Principle of 
Mathematical Induction in section 3 which has been explained clearly with 
the help of examples. In section 4 we have explained the statement of Well 
Ordering Principle and its equivalence with Principle of Mathematical 
Induction with the help of the proof. Also we have explained the basic 
properties of integers like divisibility, gcd , lcm ,Euclidean Algorithm etc  with 
the help of suitable examples. The concept of division Algorithm and 
Fundamental Theorem of Arithmetic have been explained in section 5 and 6 
respectively. 
3. Principle of Mathematical Induction(PMI): 
For proving many theorems, statements or a formula that are not proved 
directly, the Principle of Mathematical Induction is very handy and strong 
mathematical tool. The term Mathematical Induction was coined by Augustus 
Page 4


Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths 
Lesson: Principle of Mathematical Induction and Well 
Ordering Principle 
Course Developer: Dr. Roopesh Tehri 
College: Acharya Narendra Dev College, (D.U.) 
  
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
 
 
 
 
 
Table of Contents 
 Chapter : Principle of Mathematical Induction 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Principle of Mathematical Induction 
o 3.1. First Form of Principle of Mathematical Induction 
o 3.2. Second Form of Principle of Mathematical Induction 
? 4. Well Ordering Principle 
o 4.1. Equivalence of Principle of Mathematical Induction 
(PMI) and Well Ordering Principle (WOP) 
o 4.2. Euclidean Algorithm 
? 5. Division Algorithm 
? 6. Fundamental Theorem of Arithmetic 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
1. Learning Outcomes: 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
After studying the whole contents of this chapter, students will be able to 
understand:  
? The two forms of PMI with the help of suitable examples. 
? The statement of Well Ordering Principle and its equivalence with PMI. 
? The concept of basic definitions like divisibility, gcd , lcm ,Euclidean 
Algorithm etc  in integers with the help of suitable examples. 
? The concept of division Algorithm. 
? Fundamental Theorem of Arithmetic with its proof.  
 
2. Introduction: 
For future understanding of the abstract algebra, the properties of integers 
are very much handy and useful. In this lesson, we studied the basic 
properties of integers, starting with the two forms of Principle of 
Mathematical Induction in section 3 which has been explained clearly with 
the help of examples. In section 4 we have explained the statement of Well 
Ordering Principle and its equivalence with Principle of Mathematical 
Induction with the help of the proof. Also we have explained the basic 
properties of integers like divisibility, gcd , lcm ,Euclidean Algorithm etc  with 
the help of suitable examples. The concept of division Algorithm and 
Fundamental Theorem of Arithmetic have been explained in section 5 and 6 
respectively. 
3. Principle of Mathematical Induction(PMI): 
For proving many theorems, statements or a formula that are not proved 
directly, the Principle of Mathematical Induction is very handy and strong 
mathematical tool. The term Mathematical Induction was coined by Augustus 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
De Morgan.  Basically there are two forms of Principle of Mathematical 
Induction. 
3.1. First form of Principle of Mathematical Induction: 
This Form of Induction Principle says: 
If M(n) is a statement involving the positive integers n such that 
(i) If M(1) is true.  
(ii) If M(k+1) is true whenever M(k) is true.  
Then M(n) is true for all positive integers n.  
 
3.2. Second form of Principle of Mathematical Induction: 
This Form of Induction Principle says that: 
If M(n) is a statement involving the positive integers n such that 
(i) If M(1) is true, and  
(ii) Truth of  M(1), M(2), - - -, M(k) implies the truth of M(k+1).   
Then M(n) is true for all n =1.  
Value Addition: Do you know? 
Both form of the induction principle are equivalent statements. However the 
only difference between the two forms is the induction hypothesis: the first 
form assumes that M(1) is true whereas the second version assumes that all 
of M(1), M(2), ..., M(k) are true. 
Caution: The above two statements clearly indicates that Just proving 
M(k+1) whenever M(k)  is true will not work. 
 
Page 5


Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths 
Lesson: Principle of Mathematical Induction and Well 
Ordering Principle 
Course Developer: Dr. Roopesh Tehri 
College: Acharya Narendra Dev College, (D.U.) 
  
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 2 
 
 
 
 
 
 
 
Table of Contents 
 Chapter : Principle of Mathematical Induction 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Principle of Mathematical Induction 
o 3.1. First Form of Principle of Mathematical Induction 
o 3.2. Second Form of Principle of Mathematical Induction 
? 4. Well Ordering Principle 
o 4.1. Equivalence of Principle of Mathematical Induction 
(PMI) and Well Ordering Principle (WOP) 
o 4.2. Euclidean Algorithm 
? 5. Division Algorithm 
? 6. Fundamental Theorem of Arithmetic 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
1. Learning Outcomes: 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 3 
 
After studying the whole contents of this chapter, students will be able to 
understand:  
? The two forms of PMI with the help of suitable examples. 
? The statement of Well Ordering Principle and its equivalence with PMI. 
? The concept of basic definitions like divisibility, gcd , lcm ,Euclidean 
Algorithm etc  in integers with the help of suitable examples. 
? The concept of division Algorithm. 
? Fundamental Theorem of Arithmetic with its proof.  
 
2. Introduction: 
For future understanding of the abstract algebra, the properties of integers 
are very much handy and useful. In this lesson, we studied the basic 
properties of integers, starting with the two forms of Principle of 
Mathematical Induction in section 3 which has been explained clearly with 
the help of examples. In section 4 we have explained the statement of Well 
Ordering Principle and its equivalence with Principle of Mathematical 
Induction with the help of the proof. Also we have explained the basic 
properties of integers like divisibility, gcd , lcm ,Euclidean Algorithm etc  with 
the help of suitable examples. The concept of division Algorithm and 
Fundamental Theorem of Arithmetic have been explained in section 5 and 6 
respectively. 
3. Principle of Mathematical Induction(PMI): 
For proving many theorems, statements or a formula that are not proved 
directly, the Principle of Mathematical Induction is very handy and strong 
mathematical tool. The term Mathematical Induction was coined by Augustus 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 4 
 
De Morgan.  Basically there are two forms of Principle of Mathematical 
Induction. 
3.1. First form of Principle of Mathematical Induction: 
This Form of Induction Principle says: 
If M(n) is a statement involving the positive integers n such that 
(i) If M(1) is true.  
(ii) If M(k+1) is true whenever M(k) is true.  
Then M(n) is true for all positive integers n.  
 
3.2. Second form of Principle of Mathematical Induction: 
This Form of Induction Principle says that: 
If M(n) is a statement involving the positive integers n such that 
(i) If M(1) is true, and  
(ii) Truth of  M(1), M(2), - - -, M(k) implies the truth of M(k+1).   
Then M(n) is true for all n =1.  
Value Addition: Do you know? 
Both form of the induction principle are equivalent statements. However the 
only difference between the two forms is the induction hypothesis: the first 
form assumes that M(1) is true whereas the second version assumes that all 
of M(1), M(2), ..., M(k) are true. 
Caution: The above two statements clearly indicates that Just proving 
M(k+1) whenever M(k)  is true will not work. 
 
Principle of Mathematical Induction and Well Ordering Principle 
Institute of Lifelong Learning, University of Delhi                                                      pg. 5 
 
A slight generalization of PMI is as follows: 
“If a sequence of statements A
s
, A
s+1
, A
s+2
, . . . is given, where s is some 
positive integer, and if  
(i) for every value r = s, the truth of A
r+1
 will follow from the truth 
 of A
r
, i.e., A
r
 is true ? A
r+1
 is true for all r = s , and 
(ii) A
s
 is known to be true, 
then all the statements A
s
, A
s+1
, A
s+2
, . . . are true, i.e, A
n
 is true for all n = 
s." 
Example 3.1: Using PMI method show that  
                          1+ 3 + 5 + 7 + . . . + (2n-1) = n
2
  
Solution 3.1: Let M (n): 1+ 3 + 5 + 7 + . . . + (2n-1) = n
2 
                   (1) 
Putting n=1 in (1) we get 1=1
2
 which is true, hence M (1) is true. 
Now, let us assume M (k) is true. Putting n = k in (1) we have  
 M (k): 1+ 3 + 5 + 7 + . . . + (2k-1) = k
2 
                                      (2) 
Now we have to prove that M (k+1) is also true, that is 
 M (k+1): 1+ 3 + 5 + 7 + . . . + (2k-1) + (2(k+1)-1) = (k+1)
2 
      (3)  
i.e 
 M (k+1): 1+ 3 + 5 + 7 + . . . + (2k-1) + (2k+1) = (k+1)
2 
            (4)  
L.H.S of (4) = 1+ 3 + 5 + 7 + . . . + (2k-1) + (2k+1) 
                   = k
2 
+ (2k+1)   using (2) 
                   = (k+1)
2 
    
                   = R.H.S of (4)