1. Consider the result: For all integers x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. (a) Write a paragraph proof of this result using the proof of contrapositive technique. (b) Write a paragraph proof of this result using the proof by contradiction technique. 2. Construct a proof by contradiction to prove the result: No odd integer can be expressed as the sum of three even integers. 3. Suppose that you would like to prove the following implication: "For all numbers n, if n is prime then n is solitary." Write out the assumption you would state at the beginning of the argument if you were to prove the statement, (a) Directly (b) By contrapositive (c) By contradiction Note: You do not need to actually prove the result (since you do not know what solitary means).
1. Consider the result: For all integers x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. (a) Write a paragraph proof of this result using the proof of contrapositive technique. (b) Write a paragraph proof of this result using the proof by contradiction technique. 2. Construct a proof by contradiction to prove the result: No odd integer can be expressed as the sum of three even integers. 3. Suppose that you would like to prove the following implication: "For all numbers n, if n is prime then n is solitary." Write out the assumption you would state at the beginning of the argument if you were to prove the statement, (a) Directly (b) By contrapositive (c) By contradiction Note: You do not need to actually prove the result (since you do not know what solitary means).
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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Please answer the following questions.
Transcribed Image Text:1. Consider the result: For all integers x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5.
(a) Write a paragraph proof of this result using the proof of contrapositive technique.
(b) Write a paragraph proof of this result using the proof by contradiction technique.
2. Construct a proof by contradiction to prove the result: No odd integer can be expressed as the sum of
three even integers.
3. Suppose that you would like to prove the following implication:
"For all numbers n, if n is prime then n is solitary."
Write out the assumption you would state at the beginning of the argument if you were to prove the
statement,
(a) Directly
(b) By contrapositive
(c) By contradiction
Note: You do not need to actually prove the result (since you do not know what solitary means).
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