Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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 \( y \).
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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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