(d) Prove the following theorem using the contrapositive. Theorem. Suppose x and y are real numbers, and x +y is irrational. Then at least one of x and y is irrational. You should justify any general properties of rational numbers you use. (e) The following theorem is true, but the proof is wrong. Explain where the mistake is in the proof. Write no more than four sentences. Theorem. Suppose ne N. Then 23-1 is divisible by 7. Proof. 23n = (23) = 8", which is divisible by 8 because n > 0. So 23n is a multiple of 8, and therefore 23-1 is a multiple of 8-1, i.e. is a multiple of 7.
(d) Prove the following theorem using the contrapositive. Theorem. Suppose x and y are real numbers, and x +y is irrational. Then at least one of x and y is irrational. You should justify any general properties of rational numbers you use. (e) The following theorem is true, but the proof is wrong. Explain where the mistake is in the proof. Write no more than four sentences. Theorem. Suppose ne N. Then 23-1 is divisible by 7. Proof. 23n = (23) = 8", which is divisible by 8 because n > 0. So 23n is a multiple of 8, and therefore 23-1 is a multiple of 8-1, i.e. is a multiple of 7.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer question d and E please
![11:34
(a) Suppose ne N. Define what is meant by the factorial n!.
(b) Evaluate
1001 - 1000! x 999
999!
[You do not have to show your working, but doing so may help you to gain marks if you make
arithmetic errors.]
(c) Suppose P, Q and A are statements. Complete the following truth table for the statement
"(PQ) and ((not Q) or R)".
P Q R (PQ) and ((not Q) or R)
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF
?
F
?
F
T
?
T
?
[Don't copy the whole table - just write the four missing entries in order from top to bottom in
your answer booklet. You don't need to show any working.]
(d) Prove the following theorem using the contrapositive.
Theorem. Suppose x and y are real numbers, and x + y is irrational. Then at least one of x
and y is irrational.
You should justify any general properties of rational numbers you use.
(e) The following theorem is true, but the proof is wrong. Explain where the mistake is in the
proof. Write no more than four sentences.
Theorem. Suppose ne N. Then 23-1 is divisible by 7.
Proof. 23 (23) = 8", which is divisible by 8 because n > 0. So 230 is a multiple of 8, and
therefore 23 - 1 is a multiple of 8-1, i.e. is a multiple of 7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ec59470-7284-445f-830c-21983d8e4927%2Fb49e5623-5c20-4cf0-9ae7-36cd930f379e%2Fjzk4nrr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11:34
(a) Suppose ne N. Define what is meant by the factorial n!.
(b) Evaluate
1001 - 1000! x 999
999!
[You do not have to show your working, but doing so may help you to gain marks if you make
arithmetic errors.]
(c) Suppose P, Q and A are statements. Complete the following truth table for the statement
"(PQ) and ((not Q) or R)".
P Q R (PQ) and ((not Q) or R)
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF
?
F
?
F
T
?
T
?
[Don't copy the whole table - just write the four missing entries in order from top to bottom in
your answer booklet. You don't need to show any working.]
(d) Prove the following theorem using the contrapositive.
Theorem. Suppose x and y are real numbers, and x + y is irrational. Then at least one of x
and y is irrational.
You should justify any general properties of rational numbers you use.
(e) The following theorem is true, but the proof is wrong. Explain where the mistake is in the
proof. Write no more than four sentences.
Theorem. Suppose ne N. Then 23-1 is divisible by 7.
Proof. 23 (23) = 8", which is divisible by 8 because n > 0. So 230 is a multiple of 8, and
therefore 23 - 1 is a multiple of 8-1, i.e. is a multiple of 7.
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