Question

Let $\mathit{Q}\left(x,y\right)$be the statement “xhas sent an e-mail message to y,” where the domain for both xand yconsists 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)$\mathbf{\forall }\mathit{x}\mathbf{\forall }\mathit{y}\mathit{Q}\left(x,y\right)$

Short answer

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.”

Step-by-step solution

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.

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.