Question

Consider the FOL sentence F: ∀x(∃y R(x, y)). Assuming non-empty logical domains, which of the sentences below are implied by F?
(A) ∃y(∃x R(x, y))
(B) ∃y(∀x R(x, y))
(C) ∀y(∃x R(x, y))
(D) ¬∃x(∀y ¬R(x, y))
Choose the correct answer from the options given below:

• (C) and (D) only
• (A) and (D) only
• (A) and (B) only
• (B) and (C) only

Hint:

In First-Order Logic, carefully examine the quantifiers (∀, ∃) and their order. Changing the order of quantifiers can significantly alter the meaning of a statement.

Answer 1

The Correct Option is A

F : ∀x(∃y R(x, y)) means that for every x, there exists a y such that the relation R(x, y) holds. This implies that for every possible value of x, at least one y satisfies the condition R(x, y). ∀y(∃x R(x, y)) suggests that for every y, there exists an x such that R(x, y) holds. ¬∃x(∀y ¬R(x, y)) suggests that there does not exist an x for which R(x, y) fails for all y.