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Found in: Page 64

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Let ${\mathbit{Q}}\left(x,y\right)$ 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) ${\mathbf{\exists }}{\mathbit{x}}{\mathbf{\exists }}{\mathbit{y}}{\mathbf{Q}}\left(x,y\right)$ (b) ${\mathbf{\exists }}{\mathbit{x}}{\mathbf{\forall }}{\mathbit{y}}{\mathbit{Q}}\left(x,y\right)$(c) ${\mathbf{\forall }}{\mathbit{x}}{\mathbf{\exists }}{\mathbit{y}}{\mathbit{Q}}\left(x,y\right)$ (d) ${\mathbf{\exists }}{\mathbit{y}}{\mathbf{\forall }}{\mathbit{x}}{\mathbit{Q}}\left(x,y\right)$ (e) ${\mathbf{\forall }}{\mathbit{y}}{\mathbf{\exists }}{\mathbit{x}}{\mathbit{Q}}\left(x,y\right)$ (f) ${\mathbf{\forall }}{\mathbit{x}}{\mathbf{\forall }}{\mathbit{y}}{\mathbit{Q}}\left(x,y\right)$

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.”

See the step by step solution

## 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) $\exists x\exists y\mathrm{Q}\left(x,y\right)$

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

(b) $\exists x\forall yQ\left(x,y\right)$

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

(c) $\forall x\exists yQ\left(x,y\right)$

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

(d) $\exists y\forall xQ\left(x,y\right)$

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

(e) $\forall y\exists xQ\left(x,y\right)$

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

(f) $\forall x\forall yQ\left(x,y\right)$

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.