Think of a proof as a way to communicate a convincing argument for the truth of a mathematical statement. When you write a proof, imagine that you will be sending it to a capable classmate who has had to miss the last week or two of your course. Try to be clear and complete. Keep in mind that your classmate will see only what you actually write down, not any unexpressed thoughts behind it. Ideally, your proof will lead your classmate to understand why the given statement is true. Theorem 4.1.1 The sum of any two even integers is even. Proof: Suppose m and n are [particular but arbitrarily chosen] even integers. [We must show that m+n is even.] By definition of even, m = 2r and n = 2s for some integers r and s. Then m+n=2r+2s = 2(r+s) by substitution by factoring out a 2. Lettr+s. Note that t is an integer because it is a sum of integers. Hence m+n=21 where t is an integer. It follows by definition of even that m+n is even. [This is what we needed to show.] In case the statement is false, determine whether a small change would make it true. If so, make the change and prove the new statement. 19. For all real numbers a and b, if a < b then a < atb
Think of a proof as a way to communicate a convincing argument for the truth of a mathematical statement. When you write a proof, imagine that you will be sending it to a capable classmate who has had to miss the last week or two of your course. Try to be clear and complete. Keep in mind that your classmate will see only what you actually write down, not any unexpressed thoughts behind it. Ideally, your proof will lead your classmate to understand why the given statement is true. Theorem 4.1.1 The sum of any two even integers is even. Proof: Suppose m and n are [particular but arbitrarily chosen] even integers. [We must show that m+n is even.] By definition of even, m = 2r and n = 2s for some integers r and s. Then m+n=2r+2s = 2(r+s) by substitution by factoring out a 2. Lettr+s. Note that t is an integer because it is a sum of integers. Hence m+n=21 where t is an integer. It follows by definition of even that m+n is even. [This is what we needed to show.] In case the statement is false, determine whether a small change would make it true. If so, make the change and prove the new statement. 19. For all real numbers a and b, if a < b then a < atb
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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![Directions for Writing Proofs of Universal Statements:
Think of a proof as a way to communicate a convincing argument for the truth of a mathematical
statement. When you write a proof, imagine that you will be sending it to a capable classmate who has
had to miss the last week or two of your course. Try to be clear and complete. Keep in mind that your
classmate will see only what you actually write down, not any unexpressed thoughts behind it. Ideally,
your proof will lead your classmate to understand why the given statement is true.
Theorem 4.1.1
The sum of any two even integers is even.
Proof:
Suppose m and n are [particular but arbitrarily chosen] even integers. [We must show
that m +n is even.] By definition of even, m = 2r and n = 2s for some integers r
and s. Then
m+n=2r +2s by substitution
= 2(r+s)
by factoring out a 2.
Let t = r + s. Note that is an integer because it is a sum of integers. Hence
m + n = 2t where t is an integer.
It follows by definition of even that m +n is even. [This is what we needed to show.]*
In case the statement is false, determine whether a small change would make it true. If so, make the
change and prove the new statement.
19. For all real numbers a and b, if a < b then a <
a+b
<b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1dbff557-c987-45fc-8dd6-6084416c9157%2F6969613b-fc4a-49cd-b25e-87f413ee6b77%2F0uj4nwt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Directions for Writing Proofs of Universal Statements:
Think of a proof as a way to communicate a convincing argument for the truth of a mathematical
statement. When you write a proof, imagine that you will be sending it to a capable classmate who has
had to miss the last week or two of your course. Try to be clear and complete. Keep in mind that your
classmate will see only what you actually write down, not any unexpressed thoughts behind it. Ideally,
your proof will lead your classmate to understand why the given statement is true.
Theorem 4.1.1
The sum of any two even integers is even.
Proof:
Suppose m and n are [particular but arbitrarily chosen] even integers. [We must show
that m +n is even.] By definition of even, m = 2r and n = 2s for some integers r
and s. Then
m+n=2r +2s by substitution
= 2(r+s)
by factoring out a 2.
Let t = r + s. Note that is an integer because it is a sum of integers. Hence
m + n = 2t where t is an integer.
It follows by definition of even that m +n is even. [This is what we needed to show.]*
In case the statement is false, determine whether a small change would make it true. If so, make the
change and prove the new statement.
19. For all real numbers a and b, if a < b then a <
a+b
<b.
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