Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Let $Q (x, y)$ be the statement “x has sent an e-mail message to y,” where the domain for both x and y consists of all students in your class. Express each of these quantifications in English.
(a) $\exists x \exists y Q (x, y)$ (b) $\exists x \forall y Q (x, y)$
(c) $\forall x \exists y Q (x, y)$ (d) $\exists y \forall x Q (x, y)$
(e) $\forall y \exists x Q (x, y)$ (f) $\forall x \forall y Q (x, y)$

For expressing the given statements in English, use the significance of quantifiers. Here, the quantifier “ $\forall$ ” indicates “All” whereas the quantifier “ $\exists$ ” represents “Some” or “There exists.”

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Quantifier

Quantifiers are terms that correspond to quantities such as "some" or "all" and indicate the number of items for which a certain proposition is true.

Step 2: Translation of statements into English

(a) $\exists x \exists y Q (x, y)$

This indicates that there is a student in your class who has sent a message to some student in your class.

(b) $\exists x \forall y Q (x, y)$

This indicates that there is a student in your class who has sent a message to all students in your class.

This indicates that all students in your class have sent a message to at least one student in your class.

(d) $\exists y \forall x Q (x, y)$

This indicates that there is a student in your class who has been sent a message by every student in your class.

(e) $\forall y \exists x Q (x, y)$

This indicates that every student in your class has been sent a message from at least one student in your class.

(f) $\forall x \forall y Q (x, y)$

This indicates that every student in your class has sent a message to every student in the class.

Therefore, the given statements have been expressed in English.

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Discrete Mathematics and its Applications

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