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
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13912691-4f2a-48b8-8b2a-dac872326113%2F69d85e27-6659-4601-92a2-46e5850fefd2%2F68k23s_processed.jpeg&w=3840&q=75)
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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