Explanation Given information: The integers n and m are given as n ≥ 1 and m ≥ 1 . Proof: First, let’s consider when m = 1 , → n m = 1 = n 1 = n Because n ≥ 1 is a given condition, n m ≥ 1 is true for m = 1 . Then, let’s assume that the result is true for m = p where p ≥ 1 . → n m = p = n p ≥ 1 Now, for m = p + 1 , → n m = p + 1 To determine (b) To prove: That n r ≤ n s for all integer r and s where r ≤ s for any integer n ≥ 1 .

5th Edition

ISBN: 9780357540244

Author: EPP

Publisher: CENGAGE L

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Chapter 11.2, Problem 46ES

To determine

(a)

To prove:

That nm≥1 for every integer m≥1, for any integer n≥1, using the mathematical induction.

To determine

(b)

To prove:

That nr≤ns for all integer r and s where r≤s for any integer n≥1.

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Chapter 11.2, Problem 45ES

Chapter 11.2, Problem 47ES

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(a) To prove: That n m ≥ 1 for every integer m ≥ 1 , for any integer n ≥ 1 , using the mathematical induction.

Solution Summary: The author explains how to prove nmge 1 for every integer n and m using the mathematical induction.

Concept explainers

(a)

To prove:

That nm≥1 for every integer m≥1, for any integer n≥1, using the mathematical induction.

(b)

To prove:

That nr≤ns for all integer r and s where r≤s for any integer n≥1.

Want to see the full answer?

Chapter 11 Solutions

(a) To prove: That n m ≥ 1 for every integer m ≥ 1 , for any integer n ≥ 1 , using the mathematical induction.

Solution Summary: The author explains how to prove nmge 1 for every integer n and m using the mathematical induction.

Concept explainers

(a) To prove: That nm≥1 for every integer m≥1, for any integer n≥1, using the mathematical induction.

(b) To prove: That nr≤ns for all integer r and s where r≤s for any integer n≥1.

Want to see the full answer?

Chapter 11 Solutions

(a)

To prove:

That nm≥1 for every integer m≥1, for any integer n≥1, using the mathematical induction.

(b)

To prove:

That nr≤ns for all integer r and s where r≤s for any integer n≥1.