Question

Let W(x,y)mean that studentx has visitedwebsite y, where the domain for xconsists of all students in your school and the domain for yconsists of all websites. Express each of these statements by a simple Englishsentence.

(a)$\mathbf{W}\mathbf{(}\mathbf{SarahSmith}\mathbf{,}\mathbf{www}\mathbf{.}\mathbf{att}\mathbf{.}\mathbf{com}\mathbf{)}$

(b)$\mathbf{\exists }\mathbf{xW}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{www}\mathbf{.}\mathbf{imdb}\mathbf{.}\mathbf{org}\mathbf{)}$

(c) $\mathbf{\exists }\mathbf{yW}\mathbf{(}\mathbf{Jose}\mathbf{\text{'}}\mathbf{Orez}\mathbf{,}\mathbf{y}\mathbf{)}$

(d)$\mathbf{\exists }\mathbf{y}\mathbf{\left(}\mathbf{W}\mathbf{\right(}\mathbf{Ashokpuri}\mathbf{,}\mathbf{y}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{W}\mathbf{(}\mathbf{cindyYoon}\mathbf{,}\mathbf{y}\mathbf{)}$

(e)$\mathbf{\exists }\mathbf{y}\mathbf{\forall }\mathbf{z}\mathbf{(}\mathbf{y}\mathbf{\ne }\mathbf{(}\mathbf{DavidBelcher}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{W}\mathbf{(}\mathbf{DavidBelcher}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\to }\mathbf{W}\mathbf{(}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

(f)$\mathbf{\exists }\mathbf{x}\mathbf{\exists }\mathbf{y}\mathbf{\forall }\mathbf{z}\mathbf{\left(}\mathbf{\right(}\mathbf{x}\mathbf{\ne }\mathbf{y}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{W}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\leftrightarrow }\mathbf{W}\mathbf{(}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

Short answer

Forexpressing 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) $\mathbf{W}\mathbf{(}\mathbf{SarahSmith}\mathbf{,}\mathbf{www}\mathbf{.}\mathbf{att}\mathbf{.}\mathbf{com}\mathbf{)}$

This indicates that student Sarah Smith has visited the website www.att.com

(b)$\mathbf{\exists }\mathbf{xW}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{www}\mathbf{.}\mathbf{imdb}\mathbf{.}\mathbf{org}\mathbf{)}$

This indicates that there is a student in your school that has visited the website www.imbd.org.

(c) $\mathbf{\exists }\mathbf{yW}\mathbf{(}\mathbf{Jose}\mathbf{\text{'}}\mathbf{Orez}\mathbf{,}\mathbf{y}\mathbf{)}$

This indicates that there is a website that Jos’e Orez has visited.

(d) $\mathbf{\exists }\mathbf{y}\mathbf{\left(}\mathbf{W}\mathbf{\right(}\mathbf{Ashokpuri}\mathbf{,}\mathbf{y}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{W}\mathbf{(}\mathbf{cindyYoon}\mathbf{,}\mathbf{y}\mathbf{)}$

This indicates that there is a website that Ashok Puri and Cindy Yoon have both visited.

(e)$\mathbf{\exists }\mathbf{y}\mathbf{\forall }\mathbf{z}\mathbf{(}\mathbf{y}\mathbf{\ne }\mathbf{(}\mathbf{DavidBelcher}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{W}\mathbf{(}\mathbf{DavidBelcher}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\to }\mathbf{W}\mathbf{(}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

This indicates that there is a student in your school, besides David Belcher, that has visited all websites that David Belcher visited.

(f) $\mathbf{\exists }\mathbf{x}\mathbf{\exists }\mathbf{y}\mathbf{\forall }\mathbf{z}\mathbf{\left(}\mathbf{\right(}\mathbf{x}\mathbf{\ne }\mathbf{y}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{W}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\leftrightarrow }\mathbf{W}\mathbf{(}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

This indicates that there are two different students in your class that has visited the same websites.

Therefore, the given statements have been expressed in simple English sentences.