Question

Let Q(x,y)be the statement “studenthas xbeen a contestant on quiz show y.” Express each of these sentences in terms of Q(x,y), quantifiers, and logical connectives, where the domain for xconsists of all students at your school and for yconsists of all quiz shows on television.

(a)There is a student at your school who has been a contestant on a televisionquiz show.

(b) No student at your school who has ever been a contestant on a television quiz show.

(c) There is a student at your school who has been a contestant on Jeopardyand on Wheel of Fortune.

(d) Every television quiz show has had a student from your school as a contestant.

(e) At least two students from your school have been contestants on Jeopardy.

Short answer

Forexpressing the given statements in quantifiers and logical connectives, understandthe meaning of statements and use properquantifier and logical connectives for them. Here, the quantifier “$\mathbf{\forall }$” indicates “All” whereas the quantifier “$\mathbf{\exists }$” 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)There is a student at your school who has been a contestant on a television quiz show.

Here, “There is” indicates the quantifier “$\exists$”. The symbolic representation of above statement is $\mathbf{\exists }\mathbf{x}\mathbf{\exists }\mathbf{yQ}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(b) No student at your school who has ever been a contestant on a television quiz show.

In simple words, every student at your school has not been a contestant on a quiz show. Here, “Every” indicates the quantifier “$\mathbf{\forall }$”. The symbolic representation of above statement is $\mathbf{\forall }\mathbf{x}\mathbf{\forall }\mathbf{y}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(c) There is a student at your school who has been a contestant on Jeopardy and on Wheel of Fortune.

Here, “There is” indicates the quantifier “$\exists$” and “and” represents the logical connective “$\mathbf{\wedge }$”. The symbolic representation of above statement is $\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{Q}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{Jeopardy}\mathbf{)}\mathbf{\wedge }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Wheeloffortune}\mathbf{\left)}\mathbf{\right)}$.

(d) Every television quiz show has had a student from your school as a contestant.

Here, “Every” indicates the quantifier “$\forall$”. The symbolic representation of above statement is $\mathbf{\forall }\mathbf{y}\mathbf{\exists }\mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(e) At least two students from your school have been contestants on Jeopardy.

In this statement, at least two means that there exist two different students. So, “There exist” indicates the quantifier “$\exists$”. The symbolic representation of above statement is $\mathbf{\exists }\mathbf{x}\mathbf{\exists }\mathbf{z}\mathbf{\left(}\mathbf{\right(}\mathbf{x}\mathbf{\ne }\mathbf{z}\mathbf{)}\mathbf{\wedge }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Jeopardy}\mathbf{)}\mathbf{\wedge }\mathbf{Q}\mathbf{(}\mathbf{z}\mathbf{,}\mathbf{Jeopardy}\mathbf{\left)}\mathbf{\right)}$.

Therefore, the given statements have been expressed in terms of quantifiers, logical connectives and Q(x,y).