The question involves using logical quantifiers to express the statements "Many people love A" and "A loves many people" and their negations. The formulas are ∃x (p(x, A)) and ∃y (p(A, y)) for the original statements, and the negations are ¬∃x (p(x, A)) and ¬∃y (p(A, y)).
The question involves expressing statements about relationships using logical quantifiers and then finding their negations. Given S as the domain of humans and p(x, y) representing the statement "x loves y", we can write the formulas for the following statements:
a) Many people love A: ∃x (p(x, A))
b) A loves many people: ∃y (p(A, y))
The negation of these statements can be written as:
a) It is not the case that many people love A: ¬∃x (p(x, A)), which means no one loves A or everyone does not love A.
b) A does not love many people: ¬∃y (p(A, y)), implying A loves no one or A does not love everyone.