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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Transcribed Image Text:### Mathematical Proof Exercises
1. **Consider the result**: For all integers \( x \) and \( y \), if \( x + y \geq 9 \), then either \( x \geq 5 \) or \( y \geq 5 \).
- (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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