"Proof": Take any group of n cats, where n = 1, 2, ... We need to prove that all cats in this group have the same color. This is an obvious statement for n = 1. Inductive step: Assume that the statement holds for k. We need to prove it for k + 1. Consider a group of k + 1 cats. Label them by numbers from 1 to k+1. By the inductive assumption, cats 1, ..., k have the same color. Similarly, by the inductive assumption cats 2, ..., k +1 have the same color, since there are k of them. Since these two sets intersect, all k + 1 cats have the same color. Where is the error in this proof?
"Proof": Take any group of n cats, where n = 1, 2, ... We need to prove that all cats in this group have the same color. This is an obvious statement for n = 1. Inductive step: Assume that the statement holds for k. We need to prove it for k + 1. Consider a group of k + 1 cats. Label them by numbers from 1 to k+1. By the inductive assumption, cats 1, ..., k have the same color. Similarly, by the inductive assumption cats 2, ..., k +1 have the same color, since there are k of them. Since these two sets intersect, all k + 1 cats have the same color. Where is the error in this proof?
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 43E
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Transcribed Image Text:Section 9 of the notes contains a "proof" of all
cats having the same color. Explain the
mistake in this argument.
Transcribed Image Text:"Proof": Take any group of n cats, where n = 1,2,... We need to prove
that all cats in this group have the same color. This is an obvious
statement for n = 1.
Inductive step: Assume that the statement holds for k. We need to
prove it for k + 1. Consider a group of k + 1 cats. Label them by
numbers from 1 to k+1. By the inductive assumption, cats 1, ..., k have
the same color. Similarly, by the inductive assumption cats 2, ..., k+1
have the same color, since there are k of them. Since these two sets
intersect, all k + 1 cats have the same color.
Where is the error in this proof?
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