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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 even integers. By definition, there are some integersp and q such that x = 2p, and y = 2q. Let k = p + q be an integer, %3D then x + y = 2p + 2q = 2(p + q) = 2k. By definition, x +y is even. Which proof method is used in above proof. O Proof by counterexample Direct proof Proof by contraposition O Proof by contradiction

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 even integers. By definition, there are some integersp and q such that x = 2p, and y = 2q. Let k = p + q be an integer, %3D then x + y = 2p + 2q = 2(p + q) = 2k. By definition, x +y is even. Which proof method is used in above proof. O Proof by counterexample Direct proof 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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**Title:** Proof That Sum of Two Even Integers is Even **Concept:** 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 even integers. By definition, there are some integers \( p \) and \( q \) such that \( x = 2p \), and \( y = 2q \). Let \( k = p + q \) be an integer, then \[ x + y = 2p + 2q = 2(p + q) = 2k. \] By definition, \( x + y \) is even. **Question:** Which proof method is used in the above proof? - ○ Proof by counterexample - ○ Direct proof - ○ Proof by contraposition - ○ Proof by contradiction
Transcribed Image Text:**Title:** Proof That Sum of Two Even Integers is Even **Concept:** 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 even integers. By definition, there are some integers \( p \) and \( q \) such that \( x = 2p \), and \( y = 2q \). Let \( k = p + q \) be an integer, then \[ x + y = 2p + 2q = 2(p + q) = 2k. \] By definition, \( x + y \) is even. **Question:** Which proof method is used in the above proof? - ○ Proof by counterexample - ○ Direct proof - ○ Proof by contraposition - ○ Proof by contradiction
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