Consider the first-order logic sentence F: ∀x (∃ y R(x, y)). Assuming no-empty logical domains, which of the sentence below are implied by F?∃y (∃x R(x, y))∃y (∀x R(x, y))∀y (∃x R(x, y))¬ ∃x (∀y ¬ R(x, y))

CONCEPT: TrueFalse∀ xAllAtleast ONE false∃ xAtleast one trueFor all x, if P(x) is false, if we are taking PREDICATE as P(x) One of the methods for this question is by considering x and y as the domains by using some statements.Let us SUPPOSE x is for a girl and y is for a boy.Given F is ∀x(∃ y R(x, y)) , in case of English LANGUAGE this means,F: All girls like some boysNow, check all the option one by one.∃y (∃x R(x, y)) means some boys are liked by some girls.From this statement it is clear that it is the subset of given statement. TRUE∃y (∀x R(x, y)) means some boys are liked by all the girls. FALSE∀y (∃x R(x, y)) means all boys are liked by some girls which is opposite of given statement. So, this is FALSE.¬ ∃x (∀y ¬ R(x, y)) means for all girls like some boys. So, this is equivalent to given statement. TRUEAlternate:¬ ∃x (∀y ¬ R(x, y)) ≡ ∀x(¬ ∀y( ¬ R(x, y))) ≡ ∀x(∃x R(x, y))∀x (∃ y R(x, y)) → ∃ x (∃ y R(x, y))