What is wrong with this “proof" that all horses are the same color? Let P(n) be the proposition that all the horses in a set of n horses are the same color. Basis Step: Clearly, P(1) is true. Inductive Step: Assume that P (k) is true, so that all the horses in any set of k horses are the same color. Consider any k +1 horses; number these as horses 1, 2, 3, .., k, k +1. Now the first k of these horses all must have the same color, and the last k of these must also have the same color. Because the set of the first k horses and the set of the last k horses overlap, all k + 1 must be the same color. This shows that P (k + 1) is true and finishes the proof by induction.
What is wrong with this “proof" that all horses are the same color? Let P(n) be the proposition that all the horses in a set of n horses are the same color. Basis Step: Clearly, P(1) is true. Inductive Step: Assume that P (k) is true, so that all the horses in any set of k horses are the same color. Consider any k +1 horses; number these as horses 1, 2, 3, .., k, k +1. Now the first k of these horses all must have the same color, and the last k of these must also have the same color. Because the set of the first k horses and the set of the last k horses overlap, all k + 1 must be the same color. This shows that P (k + 1) is true and finishes the proof by induction.
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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Transcribed Image Text:What is wrong with this “proof" that all horses are the
same color?
Let P(n) be the proposition that all the horses in a set
of n horses are the same color.
Basis Step: Clearly, P(1) is true.
Inductive Step: Assume that P (k) is true, so that all
the horses in any set of k horses are the same color.
Consider any k +1 horses; number these as horses
1, 2, 3, .., k, k +1. Now the first k of these horses all
must have the same color, and the last k of these must
also have the same color. Because the set of the first k
horses and the set of the last k horses overlap, all k + 1
must be the same color. This shows that P (k + 1) is true
and finishes the proof by induction.
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