Gatebook - predicate Logic - negating

Question

PFB practice question from GATEBOOK on predicate logic.

a) Everyone loves every one

Solution: somebody hates somebody

b) Nobody loves everybody

Solution: someone loves every body

c) Somebody loves somebody

Solution: nobody loves somebody

d) Everyone loves some one

Solution: somebody loves nobody.

is below are correct for the above statements:

a)No one Loves Everyone

b)Everybody Loves Everybody
c)everybody loves everybody

d)Nobody Loves Someone

Answer 1

Let L(x,y) represent x loves y

1. ∀x∀yL(x, y)) :Given predicate

Negation  ¬∀x∀yL(x, y) = ∃x∃y¬L(x, y) : There is someone who doesn't love someone.Somebody hates somebody

2. ∀x∃y¬L(x, y) :Given predicate says everyone hates someone

Negation  ¬∀x∃y¬L(x, y) =∃x∀yL(x, y) :- there is someone who loves everyone.

3.∃x∃yL(x, y)

Negation  ∀x∀y¬L(x, y) : Everyone hates everyone. or nobody loves somebody

4.∀x∃yL(x, y)

Negation:- .∃x∀y¬L(x, y) : There is someone who hates everyone.Or somebody loves nobody