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Which proof method is used in above proof. O Proof by counterexample O Direct proof O Proof by contraposition O Proof by contradiction

Which proof method is used in above proof. O Proof by counterexample O Direct proof O Proof by contraposition O Proof by contradiction

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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Consider the following proof that if x and y are two even integers, then their sum is even. Proof: Assume that x and y are two integers and x + y is odd. By definition, there is some integer k such that x + y = 2k + 1. If x is even, then there is some integer p such that x = 2p, and therefore, y = (x+ y) - x = (2k + 1) - 2p = 2(k - p) + 1. Since k - p is also an integer, by definition, y is odd. Similarly, if y is even, then x is odd. Putting together, we conclude that x and y cannot be both even. Which proof method is used in above proof. O Proof by counterexample Direct proof O Proof by contraposition O Proof by contradiction
Transcribed Image Text:Consider the following proof that if x and y are two even integers, then their sum is even. Proof: Assume that x and y are two integers and x + y is odd. By definition, there is some integer k such that x + y = 2k + 1. If x is even, then there is some integer p such that x = 2p, and therefore, y = (x+ y) - x = (2k + 1) - 2p = 2(k - p) + 1. Since k - p is also an integer, by definition, y is odd. Similarly, if y is even, then x is odd. Putting together, we conclude that x and y cannot be both even. Which proof method is used in above proof. O Proof by counterexample Direct proof O Proof by contraposition O Proof by contradiction
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