Question

Let F(x,y) be the statement “x can fool y,”where the domain consists of all people in the world.Use quantifiers to express each of these statements.

(a)Everybody can fool Fred.

(b) Evelyn can fool everybody.

(c) Everybody can fool somebody.

(d) There is no one who can fool everybody.

(e)Everyone can be fooled by somebody.

(f)No one can fool both Fred and Jerry.

(g)Nancy can fool exactly two people.

(h)There is exactly one person whom everybody can fool.

(i)No one can fool himself or herself.

(j)There is someone who can fool exactly one person besides himself or herself.

Short answer

Forexpressing the given statements in terms of the 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)Everybody can fool Fred.

In this statement, “everybody”means “All people in the world.”The symbolic representation of above statement is $\mathbf{\forall }\mathbf{xF}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Fred}\mathbf{)}$.

(b)Evelyn can fool everybody.

In this statement, “everybody” means “All people in the world.” The symbolic representation of above statement is $\mathbf{\forall }\mathbf{yF}\mathbf{(}\mathbf{Evelyn}\mathbf{,}\mathbf{y}\mathbf{)}$.

(c) There is somebody whom everybody loves.

In this statement, “everybody” means “All people in the world” whereas “Somebody” means “There exists a person in the world.” The symbolic representation of above statement is $\mathbf{\forall }\mathbf{x}\mathbf{\exists }\mathbf{yF}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(d) Nobody loves everybody.

In this statement, “Nobody” means, “There does not exist a person in the world” whereas “Everybody” means “All people in the world.” The symbolic representation of above statement is $\mathbf{\neg }\mathbf{\exists }\mathbf{x}\mathbf{\forall }\mathbf{yF}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(e) Everyone can be fooled by somebody.

In this statement, “everybody” means “All people in the world” whereas “Somebody” means “There exists a person in the world.” The symbolic representation of above statement is $\mathbf{\forall }\mathbf{y}\mathbf{\exists }\mathbf{xF}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.

(f) No one can fool both Fred and Jerry.

In this statement, “Nobody” means, “There does not exist a person in the world”. The symbolic representation of above statement is $\mathbf{\neg }\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{F}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{Fred}\mathbf{)}\mathbf{\wedge }\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Jerry}\mathbf{\left)}\mathbf{\right)}$.

(g) Nancy can fool exactly two people.

In simple words, Nancy can fool two peopleyandz, andyandzcannot be the same person, and all people that Nancy can fool then have to be either yorz.

The symbolic representation of above statement is $\mathbf{\exists }\mathbf{y}\mathbf{\exists }\mathbf{z}\mathbf{\left(}\mathbf{F}\mathbf{\right(}\mathbf{Nancy}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{\wedge }\mathbf{F}\mathbf{(}\mathbf{Nancy}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\wedge }\mathbf{y}\mathbf{\ne }\mathbf{z}\mathbf{\wedge }\mathbf{\forall }\mathbf{w}\mathbf{(}\mathbf{F}\mathbf{(}\mathbf{Nancy}\mathbf{,}\mathbf{w}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{w}\mathbf{=}\mathbf{y}\mathbf{\vee }\mathbf{w}\mathbf{=}\mathbf{z}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$.

(h) There is exactly one person whom everybody can fool.

In simple words, there is a person ywhom everybody can fool and all other people whom everybody can fool then have to be this persony. The symbolic representation of above statement is$\mathbf{\exists }\mathbf{y}\mathbf{(}\mathbf{\forall }\mathbf{xF}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{\wedge }\mathbf{\forall }\mathbf{z}\mathbf{(}\mathbf{\forall }\mathbf{wF}\mathbf{(}\mathbf{w}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{)}\mathbf{\to }\mathbf{z}\mathbf{=}\mathbf{y}\mathbf{)}$.

(i) No one can fool himself or herself.

In simple words, Every personxcannot foolx. The symbolic representation of above statement is $\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{x}\mathbf{)}$.

(j) There is someone who can fool exactly one person besides himself or herself.

In simple words, There is a personxfor whom there exists a person ythatxcan fool and for all other peoplez thatxcan fool, zhas to be the person yorzhas to bexhimself.The symbolic representation of above statement is $\mathbf{\exists }\mathbf{x}\mathbf{\exists }\mathbf{y}\mathbf{\left(}\mathbf{F}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{\wedge }\mathbf{\forall }\mathbf{z}\mathbf{(}\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{z}\mathbf{)}\mathbf{\to }\mathbf{z}\mathbf{=}\mathbf{y}\mathbf{\vee }\mathbf{z}\mathbf{=}\mathbf{x}\mathbf{\left)}\mathbf{\right)}$.

Therefore, the given statements have been expressed in terms of quantifiers, logical connectives and $\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$.