# SOLVEDUsing Quantifiers to express the sentences?

#### racewithferrari

Let F(x,y) be the statement "x can fool y," where domain consist of all the people in the world.
Use Quantifiers to express the statements.

a) Everyone can be fooled by somebody.

b) Bill can fool exactly two people.

c) There is exactly one person whom everybody can fool.

d) No one can fool himself or herself.

universal quantifier is "∀"
existential quantifier is "∃"

My attempts (Worried)

a) ∀x∃y F(x,y)
b) ∃x∃y(x ≠ y Λ F(x, Bill) Λ F(x, Bill) V ∀z (F(x,z) → (z = x V z = y)))
c) ∃x (∀y F(x,y) Λ ∀z((∀w F(w,z)) → z=x))
d) ¬∃x F(x,x)

Last edited:

#### HallsofIvy

MHF Helper
Let F(x,y) be the statement "x can fool y," where domain consist of all the people in the world.
Use Quantifiers to express the statements.

a) Everyone can be fooled by somebody.

b) Bill can fool exactly two people.

c) There is exactly one person whom everybody can fool.

d) No one can fool himself or herself.

universal quantifier is "∀"
existential quantifier is "∃"

My attempts (Worried)

a) ∀x∃y F(x,y)
This would say "everyone can full somebody". You want ∀y∃x F(x,y)
b) ∃x∃y(x ≠ y Λ F(x, Bill) Λ F(x, Bill) V ∀z (F(x,z) → (z = x V z = y)))
c) ∃x (∀y F(x,y) Λ ∀z((∀w F(w,z)) → z=x))
d) ¬∃x F(x,x)