Question

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?

1. ∃y (∃x R(x, y))
2. ∃y (∀x R(x, y))
3. ∀y (∃x R(x, y))
4. ¬ ∃x (∀y ¬ R(x, y))

1. IV only 
2. I and IV only
3. II only
4. II and III only 

Answer 1

Correct Answer - Option 2 : I and IV only

Concept:

	True	False
∀ x	All	Atleast one false
∃ x	Atleast one true	For 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 boys

Now, check all the option one by one.

1. ∃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

2. ∃y (∀x R(x, y)) means some boys are liked by all the girls. FALSE
3. ∀y (∃x R(x, y)) means all boys are liked by some girls which is opposite of given statement. So, this is FALSE.
4. ¬ ∃x (∀y ¬ R(x, y)) means for all girls like some boys. So, this is equivalent to given statement. TRUE

Alternate:

¬ ∃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))