Proposition: Let a and b be integers. If a is even and a divides b. then b is also even Choose a match Proof: Assume p is true, so that a and b are integers, a is even, • Proof by Contradiction and a divides b.
Proposition: Let a and b be integers. If a is even and a divides b. then b is also even Choose a match Proof: Assume p is true, so that a and b are integers, a is even, • Proof by Contradiction and a divides b.
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
![Proposition: Let a and b be integers. If a is even and a
Choose a match
divides b, then bis also even
Proof:
Assume p is true, so that a and b are integers, a is even,
Proof by Contradiction
and a divides b.
By definition, there exists an integer k with a = 2k, and
there exists an integer withb=a
By substitution, we can writeb-a = (2k) = 2(k').
Since b = 2(k )b is even.
O Proof by Contrapositive
O Direct Proof
Statement: There are no integers a. b for which 2a + 4b
Choose a match
= 1
Proof:
Suppose the proposition is false, so that there are
integers a, b for which 2a + 4b 1. Dividing both sides
of this equation by 2. we conclude that a+ 2b = 1/2.
Since a and b are integers, a + 2b is also an integer. But
1/2 is not an integer, so this is impossible.
Therefore, the proposition cannot be false, so it must be
true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd13678ad-40a6-4722-b4cf-aadfb6f0ca8e%2Fa8315f75-8e70-4ce2-9800-64e5308be885%2Fp8xc063g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Proposition: Let a and b be integers. If a is even and a
Choose a match
divides b, then bis also even
Proof:
Assume p is true, so that a and b are integers, a is even,
Proof by Contradiction
and a divides b.
By definition, there exists an integer k with a = 2k, and
there exists an integer withb=a
By substitution, we can writeb-a = (2k) = 2(k').
Since b = 2(k )b is even.
O Proof by Contrapositive
O Direct Proof
Statement: There are no integers a. b for which 2a + 4b
Choose a match
= 1
Proof:
Suppose the proposition is false, so that there are
integers a, b for which 2a + 4b 1. Dividing both sides
of this equation by 2. we conclude that a+ 2b = 1/2.
Since a and b are integers, a + 2b is also an integer. But
1/2 is not an integer, so this is impossible.
Therefore, the proposition cannot be false, so it must be
true.
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