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8. Explain why the following "proof" is incorrect.

8. Explain why the following "proof" is incorrect.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. Explain why the following "proof" is incorrect. Proposition 0.1. All dogs are the same color. Proof. We will prove this proposition using mathematical induction. For each natural number n, we set P(n): Any set of n dogs consists entirely of dogs of the same color. We will prove that for each natural number n, P(n) is true, which will prove that all dogs are the same color. Base case: A set with only one dog consists entirely of dogs of the same color and, hence, P(1) is true. Inductive step: Let k be a natural number and assume that P(k) is true, that is, that every set of k dogs consists of dogs of the same color. Now consider a set D of k+1 dogs, where D = {d1, d2,. . dk+1}. %3D If we remove the dog di from the set D, we then have a set D1 of k dogs, and using the assumption that P(k) is true, these dogs must all be of the same color. Similarly, if we remove de+1 from the D, we again have a set D2 of k dogs, and these dogs must all be the same color. Furthermore, because dogs d2, . ., dk are in both sets D1 and D2, the dogs in D1 and D2 must be the same color. Since D = D1 U D2, we have proved that all of the dogs in D must be of the same color. This proves that if P(k) is true, then P(k +1) is true and, hence, by mathematical induction, we have proved that for each natural number n, any set of n dogs consists entirely of dogs of the same color. ....
Transcribed Image Text:8. Explain why the following "proof" is incorrect. Proposition 0.1. All dogs are the same color. Proof. We will prove this proposition using mathematical induction. For each natural number n, we set P(n): Any set of n dogs consists entirely of dogs of the same color. We will prove that for each natural number n, P(n) is true, which will prove that all dogs are the same color. Base case: A set with only one dog consists entirely of dogs of the same color and, hence, P(1) is true. Inductive step: Let k be a natural number and assume that P(k) is true, that is, that every set of k dogs consists of dogs of the same color. Now consider a set D of k+1 dogs, where D = {d1, d2,. . dk+1}. %3D If we remove the dog di from the set D, we then have a set D1 of k dogs, and using the assumption that P(k) is true, these dogs must all be of the same color. Similarly, if we remove de+1 from the D, we again have a set D2 of k dogs, and these dogs must all be the same color. Furthermore, because dogs d2, . ., dk are in both sets D1 and D2, the dogs in D1 and D2 must be the same color. Since D = D1 U D2, we have proved that all of the dogs in D must be of the same color. This proves that if P(k) is true, then P(k +1) is true and, hence, by mathematical induction, we have proved that for each natural number n, any set of n dogs consists entirely of dogs of the same color. ....
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