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Here's an incorrect theorem with an incorrect proof. Theorem: All people are the same age. Proof: Let P(n) be the statement "any set of n people all have the same age". If we can prove P(n) for all n, then the theorem is true. We prove P(n) by induction. Base case: Given any one person, that person has the same age as himself/herself. So P(1) is true. Inductive step: Suppose n > 2 and P(n − 1) is true. Take a set of n people: {P1, P2, ..., Pn}. Because P(n-1) is true, people P₁, P2, ..., Pn-1 all have the same age. Also because P(n-1) is true, people P2, P3, ..., Pn all have the same age. Therefore people p₁, P2, ..., Pn all have the same age. This works for any set of n people, so P(n) is true. So P(n) is true for all n. Explain in a few sentences what's wrong with the proof.

Here's an incorrect theorem with an incorrect proof. Theorem: All people are the same age. Proof: Let P(n) be the statement "any set of n people all have the same age". If we can prove P(n) for all n, then the theorem is true. We prove P(n) by induction. Base case: Given any one person, that person has the same age as himself/herself. So P(1) is true. Inductive step: Suppose n > 2 and P(n − 1) is true. Take a set of n people: {P1, P2, ..., Pn}. Because P(n-1) is true, people P₁, P2, ..., Pn-1 all have the same age. Also because P(n-1) is true, people P2, P3, ..., Pn all have the same age. Therefore people p₁, P2, ..., Pn all have the same age. This works for any set of n people, so P(n) is true. So P(n) is true for all n. Explain in a few sentences what's wrong with the proof.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 39E
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Here's an incorrect theorem with an incorrect proof. Theorem: All people are the same age. Proof: Let P(n) be the statement "any set of n people all have the same age". If we can prove P(n) for all n, then the theorem is true. We prove P(n) by induction. Base case: Given any one person, that person has the same age as himself/herself. So P(1) is true. Inductive step: Suppose n > 2 and P(n − 1) is true. Take a set of n people: {P₁, P2, ..., Pn}. - Because P(n − 1) is true, people p₁, P2, ..., Pn-1 all have the same age. Also because P(n − 1) is true, people P2, P3, ..., Pn all have the same age. Therefore people p₁, P2, …, Pn all have the sa ne age. This works for any set of n people, so P(n) is true. So P(n) is true for all n. Explain in a few sentences what's wrong with the proof.
Transcribed Image Text:Here's an incorrect theorem with an incorrect proof. Theorem: All people are the same age. Proof: Let P(n) be the statement "any set of n people all have the same age". If we can prove P(n) for all n, then the theorem is true. We prove P(n) by induction. Base case: Given any one person, that person has the same age as himself/herself. So P(1) is true. Inductive step: Suppose n > 2 and P(n − 1) is true. Take a set of n people: {P₁, P2, ..., Pn}. - Because P(n − 1) is true, people p₁, P2, ..., Pn-1 all have the same age. Also because P(n − 1) is true, people P2, P3, ..., Pn all have the same age. Therefore people p₁, P2, …, Pn all have the sa ne age. This works for any set of n people, so P(n) is true. So P(n) is true for all n. Explain in a few sentences what's wrong with the proof.
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