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Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Suppose m and n are integers. If m and n are even, then m + n is even. "Proof." Suppose m and n are even but m + n is odd. Then, there exists an integer k such that m + n = 2k + 1. Therefore, m +n+ (-2k) = 1. The left side of the equation is even because it is the sum of even integers. However, the right side, 1, is odd. Since a number cannot be both even and odd, we have a contradiction. Therefore, m +n is even. Q.E.D.

Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Suppose m and n are integers. If m and n are even, then m + n is even. "Proof." Suppose m and n are even but m + n is odd. Then, there exists an integer k such that m + n = 2k + 1. Therefore, m +n+ (-2k) = 1. The left side of the equation is even because it is the sum of even integers. However, the right side, 1, is odd. Since a number cannot be both even and odd, we have a contradiction. Therefore, m +n is even. Q.E.D.

Advanced Engineering Mathematics
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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Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Suppose m and n are integers. If m and n are even, then m + n is even. "Proof." Suppose m and n are even but m +n is odd. Then, there exists an integer k such that m +n = 2k +1. Therefore, m +n + (-2k) = 1. The left side of the equation is even because it is the sum of even integers. However, the right side, 1, is odd. Since a number cannot be both even and odd, we have a contradiction. Therefore, m+n is even. Q.E.D.
Transcribed Image Text:Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Suppose m and n are integers. If m and n are even, then m + n is even. "Proof." Suppose m and n are even but m +n is odd. Then, there exists an integer k such that m +n = 2k +1. Therefore, m +n + (-2k) = 1. The left side of the equation is even because it is the sum of even integers. However, the right side, 1, is odd. Since a number cannot be both even and odd, we have a contradiction. Therefore, m+n is even. Q.E.D.
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