Geetha has a conjecture about integers, which is of the form

∀x(P(x) ⇒ ∃yQ(x, y)),

where P is a statement about integers, and Q is a statement about pairs of integers.

Which of the following (one or more) option(s) would imply Geetha’s conjecture?

A.

∃x(P(x) ∧ ∀yQ(x, y))

B.

∀x∀yQ(x, y)

C.

∃y∀x(P(x) ⇒ Q(x, y))

D.

∃x(P(x) ∧ ∃yQ(x, y))

Solution:

L: ∀x(P(x) ⇒ ∃yQ(x, y))

For every x if P(x) is True, then there exists some y such that Q(x,y) will be True.

Option (A): ∃x(P(x) ∧ ∀y Q(x, y))

For some x, P(x) is True and for all y Q(x,y) is True, which does not imply L.

Option (B):∀x∀yQ(x, y)

For every x and every yQ(x,y) is true which implies L.

Option (C): ∃y∀x(P(x) ⇒ Q(x, y))

These exist some y such that for every x if P(x) is True then Q(x,y) is also True which implies L.

Option (D): ∃x(P(x) ∧ ∃yQ(x, y))

There exists some x for which P(x) is True and also for some yQ(x,y) is True which cannot imply L.