# GATE | Gate IT 2007 | Question 21

Which one of these first-order logic formula is valid?
(A) ∀x(P(x) => Q(x)) => (∀xP(x) => ∀xQ(x))
(B) ∃x(P(x) ∨ Q(x)) => (∃xP(x) => ∃xQ(x))
(C) ∃x(P(x) ∧ Q(x)) (∃xP(x) ∧ ∃xQ(x))
(D) ∀x∃y P(x, y) => ∃y∀x P(x, y)

Explanation: (A) LHS->RHS
LHS: For every x (if P holds then Q holds)
RHS: If P(x) holds for all x, then Q(x) holds for all x.
(B) LHS !->RHS
LHS: An x exist for which either P(x) is true or Q(x) is true.
RHS: If an x exist for which P(x) is true then another x exist for which Q(x) is true.
(C) It is not necessary that on RHS both x are same.
LHS: There exist an x for which both P(x) and Q(x) are true.
RHS: There exist an x for which P(x) is true and there exist an x for which Q(x) is true.
(D) LHS!->RHS
LHS: For every x, there exist a y such that P(x, y) holds.
RHS: There exist a y such that for all x P(x, y) holds.

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