Question

Let C(x,y)mean that student xis enrolled in class y, where the domain for xconsists of all students in your school and the domain for yconsists of all classes being given at your school. Express each of these statements by a simple Englishsentence.

(a) $\mathbf{C}\mathbf{(}\mathbf{RandyGoldberg}\mathbf{,}\mathbf{CS}\mathbf{252}\mathbf{)}$

(b)$\mathbf{\exists }\mathbf{xC}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Math}\mathbf{695}\mathbf{)}$

(c)$\mathbf{\exists }\mathbf{yC}\mathbf{(}\mathbf{Carolsitea}\mathbf{,}\mathbf{y}\mathbf{)}$

(d)$\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{C}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{Math}\mathbf{222}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{C}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{CS}\mathbf{252}\mathbf{)}$

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

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

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) $\mathbf{C}\mathbf{(}\mathbf{RandyGoldberg}\mathbf{,}\mathbf{CS}\mathbf{252}\mathbf{)}$

This indicates that Randy Goldberg is enrolled in class CS $\mathbf{252}$

(b)$\mathbf{\exists }\mathbf{xC}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Math}\mathbf{695}\mathbf{)}$

This indicates that there exists a student $\mathbf{x}$that is enrolled in class Math $\mathbf{695}$

(c) $\mathbf{\exists }\mathbf{yC}\mathbf{(}\mathbf{Carolsitea}\mathbf{,}\mathbf{y}\mathbf{)}$

This indicates that there exists a class $\mathbf{y}$that Carol Sitea is enrolled in.

(d) $\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{C}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{Math}\mathbf{222}\mathbf{\left)}\mathbf{\right)}\mathbf{\wedge }\mathbf{C}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{CS}\mathbf{252}\mathbf{)}$

This indicates that there exists a student $\mathbf{x}$that is enrolled in class Math$\mathbf{222}$and is enrolled in class CS$\mathbf{252}$.

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

This indicates that there exists a studentxand a studentysuch that for every classz, studentxand studentyare not the same student and ifxis enrolled in classzthenyis also enrolled in classz.

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

This indicates that there exists a student xand a student ysuch that for every class z, student xand student yare not the same student and xis enrolled in class zif and only if yis enrolled in class z.

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