Assume that Vxdy P (x, y) is false and that the domalh nonempty. Which of the following must also be false? Select ALL correct answers. Note: -P means the negation of P. Grading: All or Nothing.
Assume that Vxdy P (x, y) is false and that the domalh nonempty. Which of the following must also be false? Select ALL correct answers. Note: -P means the negation of P. Grading: All or Nothing.
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:**
Assume that ∀x∃y \( P(x, y) \) is false and that the domain of discourse is nonempty. Which of the following must also be false? Select **ALL** correct answers.
Note: ~\( P \) means the negation of \( P \).
Grading: All or Nothing:
- [ ] ∃x∀y \( P(x, y) \)
- [ ] ∀x∀y \( P(x, y) \)
- [ ] ∃x∃y \( P(x, y) \)
- [ ] ∃x∀y ~\( P(x, y) \)
- [ ] ∀x∀y ~\( P(x, y) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb431a66f-4058-4e70-a20a-31b3452e77d6%2Fd3d1f7c8-9c3f-4a58-8278-20c683d9394c%2F8jzogqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Assume that ∀x∃y \( P(x, y) \) is false and that the domain of discourse is nonempty. Which of the following must also be false? Select **ALL** correct answers.
Note: ~\( P \) means the negation of \( P \).
Grading: All or Nothing:
- [ ] ∃x∀y \( P(x, y) \)
- [ ] ∀x∀y \( P(x, y) \)
- [ ] ∃x∃y \( P(x, y) \)
- [ ] ∃x∀y ~\( P(x, y) \)
- [ ] ∀x∀y ~\( P(x, y) \)
![### Big-O Notation and Asymptotic Analysis
The given mathematical expression is:
\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \]
We need to determine if this expression is \( \Theta(n) \).
**Options:**
- True
- False
The notation \( \Theta(n) \) signifies asymptotic tight bound, meaning that the function grows linearly with \( n \).
### Detailed Analysis:
1. **Floor Function \(\left\lfloor \frac{n}{5} \right\rfloor\):**
- The floor function returns the greatest integer less than or equal to \( \frac{n}{5} \).
- This component grows linearly with \( n \) since \( \frac{n}{5} \) is directly proportional to \( n \).
2. **Linear Term \( 6n \):**
- This term clearly grows linearly with \( n \).
3. **Constant Term \( + 3 \):**
- This term is constant and does not change with \( n \).
### Conclusion:
Combining all the parts together:
\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \]
- Dominated by the linear terms \( \left\lfloor \frac{n}{5} \right\rfloor \) and \( 6n \).
- The constant term \( + 3 \) does not affect the overall asymptotic behavior.
Thus, the given expression indeed grows linearly with \( n \), so it is \( \Theta(n) \).
Therefore, the correct answer is:
- True](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb431a66f-4058-4e70-a20a-31b3452e77d6%2Fd3d1f7c8-9c3f-4a58-8278-20c683d9394c%2Fsk2s18k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Big-O Notation and Asymptotic Analysis
The given mathematical expression is:
\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \]
We need to determine if this expression is \( \Theta(n) \).
**Options:**
- True
- False
The notation \( \Theta(n) \) signifies asymptotic tight bound, meaning that the function grows linearly with \( n \).
### Detailed Analysis:
1. **Floor Function \(\left\lfloor \frac{n}{5} \right\rfloor\):**
- The floor function returns the greatest integer less than or equal to \( \frac{n}{5} \).
- This component grows linearly with \( n \) since \( \frac{n}{5} \) is directly proportional to \( n \).
2. **Linear Term \( 6n \):**
- This term clearly grows linearly with \( n \).
3. **Constant Term \( + 3 \):**
- This term is constant and does not change with \( n \).
### Conclusion:
Combining all the parts together:
\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \]
- Dominated by the linear terms \( \left\lfloor \frac{n}{5} \right\rfloor \) and \( 6n \).
- The constant term \( + 3 \) does not affect the overall asymptotic behavior.
Thus, the given expression indeed grows linearly with \( n \), so it is \( \Theta(n) \).
Therefore, the correct answer is:
- True
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