Explanation Given: To explain difference between generalized principle of mathematical induction and the principle of mathematical induction Induction is a defining difference between discrete and continuous mathematics. Principle of Mathematical induction It is a mathematical proof technique used to prove a given statement about any well-ordered set. Generally, it is used to establish statements for the set of all natural numbers. First we have to consider the given statement as P ( n ) . P ( n ) , Proof of Induction is a mathematical statement involving a value n . The first step is known as Base case. By Proof of Induction, in this first step we have to prove P ( n ) is true for n = 1 . It means P ( 1 ) is true. The second step is known as Induction Hypothesis (or) Induction step. In this second step, we have to prove that, if the statement is assumed to be true for any one natural number, then it must be true for the next natural number as well. By mathematically, P ( k ) ⇒ P ( k + 1 ) , which means P ( k + 1 ) is true by using the fact that P ( k ) is true. The combination of these steps shows that P ( n ) is true for all values of n . The above steps explain the principle of mathematical induction. Principle of Generalized Mathematical Induction: It is also a mathematical proof technique used to prove a given statement about any well-ordered set. Here, it is used to establish statements for the set of all elements of X where X be a set well ordered by a relation less than ( <).First we have to consider the given statement as S ( x ) . S ( x ) is a mathematical statement about an element x of X . To prove that S ( x ) is true for all x ∈ X , it is sufficient to prove the following steps. The first step is known as Base case. In the Base case, we have to consider S ( x 0 ) is true, where x 0 is the least element of X . The second step is known as Induction Hypothesis. In the second step, we have to prove that for any x ∈ X , if S ( y ) is true for all y < x , then S ( x ) is also true. The combination of these steps shows that S ( x ) is true for all values of x . The above steps explain the principle of generalized mathematical induction...

College Algebra with Modeling & Visualization (5th Edition)

5th Edition

ISBN: 9780321826138

Author: Gary K. Rockswold

Publisher: PEARSON

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Chapter 8.5, Problem 35E

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Chapter 8.5, Problem 34E

Chapter 8.5, Problem 36E

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College Algebra with Modeling & Visualization (5th Edition)

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

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