Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the argument made. (c) Finally, provide a correct proof of the theorem above.
Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the argument made. (c) Finally, provide a correct proof of the theorem above.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Exercise 1.**
A student is asked to prove:
"If \( m \) is even and \( n \) is odd, then \( m^2 + n^2 \) is odd."
The student provides the following proof:
---
**Proof.** If \( m \) is even and \( n \) is odd, then \( m^2 + n^2 \) is odd. Since \( m \) is even, \( m = 2k \) for any integer \( k \). Since \( n \) is odd, \( n = 2k + 1 \). Note that
\[
m^2 + n^2 = (2k)^2 + (2k + 1)^2 = 8k^2 + 4k + 1
\]
So \( m^2 + n^2 \) is odd because it is an even number plus 1.
---
(a) **Identify the errors in the proof above.** There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect.
(b) **In addition to errors, provide suggestions to improve the student's proof writing.** That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the argument made.
(c) **Finally, provide a correct proof of the theorem above.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0f3171b-001d-4e71-b013-5e26782f20a2%2Fb1dd54b3-5843-4ea2-9e33-9d54830196bb%2Fgglzgx_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 1.**
A student is asked to prove:
"If \( m \) is even and \( n \) is odd, then \( m^2 + n^2 \) is odd."
The student provides the following proof:
---
**Proof.** If \( m \) is even and \( n \) is odd, then \( m^2 + n^2 \) is odd. Since \( m \) is even, \( m = 2k \) for any integer \( k \). Since \( n \) is odd, \( n = 2k + 1 \). Note that
\[
m^2 + n^2 = (2k)^2 + (2k + 1)^2 = 8k^2 + 4k + 1
\]
So \( m^2 + n^2 \) is odd because it is an even number plus 1.
---
(a) **Identify the errors in the proof above.** There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect.
(b) **In addition to errors, provide suggestions to improve the student's proof writing.** That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the argument made.
(c) **Finally, provide a correct proof of the theorem above.**
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