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Q3E

Expert-verifiedFound in: Page 64

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Let ${\mathit{Q}}{\left(x,y\right)}$ ****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) ${\mathbf{\exists}}{\mathit{x}}{\mathbf{\exists}}{\mathit{y}}{\mathbf{Q}}{\left(x,y\right)}$**** (b) ${\mathbf{\exists}}{\mathit{x}}{\mathbf{\forall}}{\mathit{y}}{\mathit{Q}}{\left(x,y\right)}$**

**(c) ${\mathbf{\forall}}{\mathit{x}}{\mathbf{\exists}}{\mathit{y}}{\mathit{Q}}{\left(x,y\right)}$** **(d) ${\mathbf{\exists}}{\mathit{y}}{\mathbf{\forall}}{\mathit{x}}{\mathit{Q}}{\left(x,y\right)}$**** **

**(e) ${\mathbf{\forall}}{\mathit{y}}{\mathbf{\exists}}{\mathit{x}}{\mathit{Q}}{\left(x,y\right)}$** **(f) ${\mathbf{\forall}}{\mathit{x}}{\mathbf{\forall}}{\mathit{y}}{\mathit{Q}}{\left(x,y\right)}$**

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.”

**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.

(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.

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