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Problem 1: Let P(x, y) denote the statement "Student x has taken class y", where the domain for x consists of all students in your class, and for y consists of all computer science courses at your school. Express each of the following quantifications in English. 1. 3r 3y P(x, y) 2. ar vy P(x, y) 3. Vx 3y P(x, y) 4. 3y Vx P(x, y) 5. Vy ar P(x, y) 6. Vx Vy P(x, y)

Problem 1: Let P(x, y) denote the statement "Student x has taken class y", where the domain for x consists of all students in your class, and for y consists of all computer science courses at your school. Express each of the following quantifications in English. 1. 3r 3y P(x, y) 2. ar vy P(x, y) 3. Vx 3y P(x, y) 4. 3y Vx P(x, y) 5. Vy ar P(x, y) 6. Vx Vy P(x, y)

Advanced Engineering Mathematics
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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**Problem 1:** Let \( P(x, y) \) denote the statement “Student \( x \) has taken class \( y \),” where the domain for \( x \) consists of all students in your class, and for \( y \) consists of all computer science courses at your school. Express each of the following quantifications in English. 1. \( \exists x \, \exists y \, P(x, y) \) 2. \( \exists x \, \forall y \, P(x, y) \) 3. \( \forall x \, \exists y \, P(x, y) \) 4. \( \exists y \, \forall x \, P(x, y) \) 5. \( \forall y \, \exists x \, P(x, y) \) 6. \( \forall x \, \forall y \, P(x, y) \) **Explanation:** 1. There exists a student \( x \) and a class \( y \) such that student \( x \) has taken class \( y \). 2. There exists a student \( x \) such that for all classes \( y \), student \( x \) has taken class \( y \). 3. For every student \( x \), there exists a class \( y \) such that student \( x \) has taken class \( y \). 4. There exists a class \( y \) such that for all students \( x \), student \( x \) has taken class \( y \). 5. For every class \( y \), there exists a student \( x \) such that student \( x \) has taken class \( y \). 6. For every student \( x \) and for every class \( y \), student \( x \) has taken class \( y \).
Transcribed Image Text:**Problem 1:** Let \( P(x, y) \) denote the statement “Student \( x \) has taken class \( y \),” where the domain for \( x \) consists of all students in your class, and for \( y \) consists of all computer science courses at your school. Express each of the following quantifications in English. 1. \( \exists x \, \exists y \, P(x, y) \) 2. \( \exists x \, \forall y \, P(x, y) \) 3. \( \forall x \, \exists y \, P(x, y) \) 4. \( \exists y \, \forall x \, P(x, y) \) 5. \( \forall y \, \exists x \, P(x, y) \) 6. \( \forall x \, \forall y \, P(x, y) \) **Explanation:** 1. There exists a student \( x \) and a class \( y \) such that student \( x \) has taken class \( y \). 2. There exists a student \( x \) such that for all classes \( y \), student \( x \) has taken class \( y \). 3. For every student \( x \), there exists a class \( y \) such that student \( x \) has taken class \( y \). 4. There exists a class \( y \) such that for all students \( x \), student \( x \) has taken class \( y \). 5. For every class \( y \), there exists a student \( x \) such that student \( x \) has taken class \( y \). 6. For every student \( x \) and for every class \( y \), student \( x \) has taken class \( y \).
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