8. [Section 3.2, Problem 19] Consider the following proposition along with the included proof of the proposition: Proposition: For all integers m and n, if mn is an even integer, then m is even or n is even. Proof. For either m or n to be even, there exists an integer k such that m = 2k or n = 2k. So if we multiply m and n, the product will contain a factor of 2 and, hence, mn will be even. Please decide if this proposition (i) is false and the proof is incorrect; (ii) is true, but the proof is not correct; (iii) is true, the proof is correct, but the proof is not written well; or (iv) is true and the proof is correct and well-written. Please explain why you believe this. If you feel the proposition is true but the proof is not good, please write a proof of your own that is better. Solution:
8. [Section 3.2, Problem 19] Consider the following proposition along with the included proof of the proposition: Proposition: For all integers m and n, if mn is an even integer, then m is even or n is even. Proof. For either m or n to be even, there exists an integer k such that m = 2k or n = 2k. So if we multiply m and n, the product will contain a factor of 2 and, hence, mn will be even. Please decide if this proposition (i) is false and the proof is incorrect; (ii) is true, but the proof is not correct; (iii) is true, the proof is correct, but the proof is not written well; or (iv) is true and the proof is correct and well-written. Please explain why you believe this. If you feel the proposition is true but the proof is not good, please write a proof of your own that is better. Solution:
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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![[Section 3.2, Problem 19] Consider the following proposition along with the included proof of the proposition:
**Proposition:** For all integers \( m \) and \( n \), if \( mn \) is an even integer, then \( m \) is even or \( n \) is even.
**Proof:** For either \( m \) or \( n \) to be even, there exists an integer \( k \) such that \( m = 2k \) or \( n = 2k \). So if we multiply \( m \) and \( n \), the product will contain a factor of 2 and, hence, \( mn \) will be even. □
Please decide if this proposition (i) is false and the proof is incorrect; (ii) is true, but the proof is not correct; (iii) is true, the proof is correct, but the proof is not written well; or (iv) is true and the proof is correct and well-written.
Please explain why you believe this. If you feel the proposition is true but the proof is not good, please write a proof of your own that is better.
**Solution:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e683aa7-5658-4929-bcab-56f031f5dc76%2F825bcc35-8cc1-45fa-9565-11c338b2bb0e%2Fs6xgpf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[Section 3.2, Problem 19] Consider the following proposition along with the included proof of the proposition:
**Proposition:** For all integers \( m \) and \( n \), if \( mn \) is an even integer, then \( m \) is even or \( n \) is even.
**Proof:** For either \( m \) or \( n \) to be even, there exists an integer \( k \) such that \( m = 2k \) or \( n = 2k \). So if we multiply \( m \) and \( n \), the product will contain a factor of 2 and, hence, \( mn \) will be even. □
Please decide if this proposition (i) is false and the proof is incorrect; (ii) is true, but the proof is not correct; (iii) is true, the proof is correct, but the proof is not written well; or (iv) is true and the proof is correct and well-written.
Please explain why you believe this. If you feel the proposition is true but the proof is not good, please write a proof of your own that is better.
**Solution:**
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