Vx P(x) is equivalent to Vx ¬P(x). O True O False
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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![**Logical Statement Analysis**
The statement to evaluate is:
\[\lnot \forall x \, P(x) \text{ is equivalent to } \exists x \, \lnot P(x).\]
Determine if the following is true or false:
- ○ True
- ○ False
**Explanation:**
The expression \(\lnot \forall x \, P(x)\) translates to "It is not true that for all \(x\), \(P(x)\) is true." This means there exists at least one \(x\) for which \(P(x)\) is not true. Therefore, \(\lnot \forall x \, P(x)\) is logically equivalent to \(\exists x \, \lnot P(x)\).
The options provided are to verify this equivalence logically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4a5d476-fa3e-48a1-88f5-ec954230f216%2Fcbe23492-5186-4c0d-be5c-3dc5e91dc7af%2Fqqc1hlf_processed.png&w=3840&q=75)
Transcribed Image Text:**Logical Statement Analysis**
The statement to evaluate is:
\[\lnot \forall x \, P(x) \text{ is equivalent to } \exists x \, \lnot P(x).\]
Determine if the following is true or false:
- ○ True
- ○ False
**Explanation:**
The expression \(\lnot \forall x \, P(x)\) translates to "It is not true that for all \(x\), \(P(x)\) is true." This means there exists at least one \(x\) for which \(P(x)\) is not true. Therefore, \(\lnot \forall x \, P(x)\) is logically equivalent to \(\exists x \, \lnot P(x)\).
The options provided are to verify this equivalence logically.
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