Question

Let $\mathit{P}\left(x,y\right)$be the statement “Student xhas taken class y,” where the domain for both xconsists of all students in your class and for yconsists of all computer science courses at your school. Express each of these quantifications in English.

(a) $\mathbf{\exists }\mathit{x}\mathbf{\exists }\mathit{y}\mathbf{P}\left(x,y\right)$ (b) $\mathbf{\exists }\mathit{x}\mathbf{\forall }\mathit{y}\mathit{P}\left(x,y\right)$

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

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

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

This indicates that there is a student in your class that has taken a computer science course.

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

This indicates that there is a student in your class that has taken all computer science courses.

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

This indicates that all students in your class have taken a computer science course.

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

This indicates that there is a computer science course that every student in your class has taken.

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

This indicates that every computer science course has been taken by some students in your class.

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

This indicates that all students in your class have taken all computer science courses.

Therefore, the given statements have been expressed in English.