Determine which of the following “proofs” are correct and which are incorrect. If a proof is correct, indicate the type and if a proof is incorrect, indicate why it is incorrect.Theorem: If a and b are even integers then a – b is an even integer. Letter g and First picture only.
Determine which of the following “proofs” are correct and which are incorrect. If a proof is correct, indicate the type and if a proof is incorrect, indicate why it is incorrect.Theorem: If a and b are even integers then a – b is an even integer. Letter g and First picture only.
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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Question
Determine which of the following “proofs” are correct and which are incorrect. If a proof is
correct, indicate the type and if a proof is incorrect, indicate why it is incorrect.Theorem: If a and b are even integers then a – b is an even integer.
Letter g and First picture only.
!["Proof 10: Suppose a and bare odd and a – b is odd. Then there exist integers k₁, k₂ such that
a=2k₁ + 1, b = 2k₂ + 1. Thus we have a - b = 2k₁ +1 − (2k₂ + 1) = 2(k₁ −k₂)
so a-b is both odd and even, a contradiction.
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Q](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86b96763-ee2d-4cf7-8375-b4d30fc6a35a%2F29ab6724-d41e-49c7-97b2-9776aeba6569%2Fb13g4pd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:"Proof 10: Suppose a and bare odd and a – b is odd. Then there exist integers k₁, k₂ such that
a=2k₁ + 1, b = 2k₂ + 1. Thus we have a - b = 2k₁ +1 − (2k₂ + 1) = 2(k₁ −k₂)
so a-b is both odd and even, a contradiction.
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of 2
G
Q
![and we are finished
g) “Proof 7¹: Suppose that a and b are both even. Then there exist integers k₁, k₂ such that
a = 2k₁ and b = 2k₂. Thus, a – b = 2k₁ – 2k₂ = 2(k −k₂) so a - b is even.
h) "Proof 8: Suppose that a – b is even. Then if a is odd we are done, so suppose that a is
Then there exist integers kỵ, k² such that a − b = 2k₁ and a = 2k₂. Thus,
b=a_(a−b) = 2k₂ - 2k, = 2(kk) so b is also even.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86b96763-ee2d-4cf7-8375-b4d30fc6a35a%2F29ab6724-d41e-49c7-97b2-9776aeba6569%2Fkkoz20d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:and we are finished
g) “Proof 7¹: Suppose that a and b are both even. Then there exist integers k₁, k₂ such that
a = 2k₁ and b = 2k₂. Thus, a – b = 2k₁ – 2k₂ = 2(k −k₂) so a - b is even.
h) "Proof 8: Suppose that a – b is even. Then if a is odd we are done, so suppose that a is
Then there exist integers kỵ, k² such that a − b = 2k₁ and a = 2k₂. Thus,
b=a_(a−b) = 2k₂ - 2k, = 2(kk) so b is also even.
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