What is wrong with the following “proof" of the "fact" that n+3 = n+7 for all values of n (besides of course that the thing it is claiming to prove is false)? Proof. Let P(n) be the statement that n + 3 = n + 7. We will prove that P(n) is true for all n e N. Assume, for induction that P(k) is true. That is, k + 3 = k + 7. We must show that P(k + 1) is true. Now since k +3 = k + 7, add 1 to both sides. This gives k + 3 +1 = k + 7 + 1. %3D
What is wrong with the following “proof" of the "fact" that n+3 = n+7 for all values of n (besides of course that the thing it is claiming to prove is false)? Proof. Let P(n) be the statement that n + 3 = n + 7. We will prove that P(n) is true for all n e N. Assume, for induction that P(k) is true. That is, k + 3 = k + 7. We must show that P(k + 1) is true. Now since k +3 = k + 7, add 1 to both sides. This gives k + 3 +1 = k + 7 + 1. %3D
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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![**Title: Flawed Proof of a Mathematical Statement**
**Description:**
In this example, we examine a flawed proof regarding the statement that \( n + 3 = n + 7 \) for all values of \( n \). Apart from the obvious fact that the claim itself is false, let's explore what's wrong with the proof provided.
**Proof Analysis:**
1. **Statement Definition:**
- Let \( P(n) \) be the proposition that \( n + 3 = n + 7 \).
2. **Proof by Induction:**
- We aim to prove that \( P(n) \) holds for all \( n \in \mathbb{N} \).
3. **Base Case:**
- Assume, for induction, that \( P(k) \) is true.
- That is, assume \( k + 3 = k + 7 \).
4. **Inductive Step:**
- We must show that \( P(k + 1) \) holds true.
- Adding 1 to both sides of \( k + 3 = k + 7 \) gives:
\[
k + 3 + 1 = k + 7 + 1
\]
- Simplifies to:
\[
(k + 1) + 3 = (k + 1) + 7
\]
- This is equivalent to the statement that \( P(k + 1) \) holds true.
5. **Conclusion:**
- According to this reasoning, by the principle of mathematical induction, \( P(n) \) is purportedly true for all \( n \in \mathbb{N} \).
**QED Error:**
- The error lies in the incorrect assumption during the inductive step where it is assumed \( k + 3 = k + 7 \) is true for any integer \( k \). This foundational assumption is false, and thus invalidates the entire proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F282904df-64d2-44b1-a59e-7b2a86d0de90%2F9820a902-bdad-4272-b75f-cf9ef9541528%2F9vxgxou_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Flawed Proof of a Mathematical Statement**
**Description:**
In this example, we examine a flawed proof regarding the statement that \( n + 3 = n + 7 \) for all values of \( n \). Apart from the obvious fact that the claim itself is false, let's explore what's wrong with the proof provided.
**Proof Analysis:**
1. **Statement Definition:**
- Let \( P(n) \) be the proposition that \( n + 3 = n + 7 \).
2. **Proof by Induction:**
- We aim to prove that \( P(n) \) holds for all \( n \in \mathbb{N} \).
3. **Base Case:**
- Assume, for induction, that \( P(k) \) is true.
- That is, assume \( k + 3 = k + 7 \).
4. **Inductive Step:**
- We must show that \( P(k + 1) \) holds true.
- Adding 1 to both sides of \( k + 3 = k + 7 \) gives:
\[
k + 3 + 1 = k + 7 + 1
\]
- Simplifies to:
\[
(k + 1) + 3 = (k + 1) + 7
\]
- This is equivalent to the statement that \( P(k + 1) \) holds true.
5. **Conclusion:**
- According to this reasoning, by the principle of mathematical induction, \( P(n) \) is purportedly true for all \( n \in \mathbb{N} \).
**QED Error:**
- The error lies in the incorrect assumption during the inductive step where it is assumed \( k + 3 = k + 7 \) is true for any integer \( k \). This foundational assumption is false, and thus invalidates the entire proof.
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