9. Translate the following statement into English, where the domain for each variable consists of all real numbers. Vx [(x < 0) Vy (y² = x)]
9. Translate the following statement into English, where the domain for each variable consists of all real numbers. Vx [(x < 0) Vy (y² = x)]
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 9: Logic and Quantifiers**
Translate the following statement into English, where the domain for each variable consists of all real numbers.
\[
\forall x \, [(x < 0) \vee \exists y \, (y^2 = x)]
\]
### Explanation
This mathematical statement involves two quantifiers: the universal quantifier \(\forall\) and the existential quantifier \(\exists\).
- **Universal Quantifier \(\forall x\)**: This indicates that the statement applies to all real numbers \(x\).
- **Existential Quantifier \(\exists y\)**: This means there exists at least one real number \(y\) such that the condition \(y^2 = x\) is true.
### Logical Translation
The formula can be expressed in English as:
"For every real number \(x\), either \(x\) is less than 0, or there exists a real number \(y\) such that \(y^2\) is equal to \(x\)."
### Further Clarification
- The part \(x < 0\) indicates that one possibility for each \(x\) is for it to be a negative number.
- The part \(\exists y \, (y^2 = x)\) indicates that if \(x\) is not negative, then \(x\) must be a perfect square of some real number \(y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F868daa41-93eb-4d05-be30-2d6217bf0561%2F6237fd09-8cfe-491f-a286-de2a28c7fa72%2Fatvqqe_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 9: Logic and Quantifiers**
Translate the following statement into English, where the domain for each variable consists of all real numbers.
\[
\forall x \, [(x < 0) \vee \exists y \, (y^2 = x)]
\]
### Explanation
This mathematical statement involves two quantifiers: the universal quantifier \(\forall\) and the existential quantifier \(\exists\).
- **Universal Quantifier \(\forall x\)**: This indicates that the statement applies to all real numbers \(x\).
- **Existential Quantifier \(\exists y\)**: This means there exists at least one real number \(y\) such that the condition \(y^2 = x\) is true.
### Logical Translation
The formula can be expressed in English as:
"For every real number \(x\), either \(x\) is less than 0, or there exists a real number \(y\) such that \(y^2\) is equal to \(x\)."
### Further Clarification
- The part \(x < 0\) indicates that one possibility for each \(x\) is for it to be a negative number.
- The part \(\exists y \, (y^2 = x)\) indicates that if \(x\) is not negative, then \(x\) must be a perfect square of some real number \(y\).
Expert Solution
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Here the domain for x and y are real numbers.
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