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)Read More

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