Answer
We want propositional functions P and Q that are sometimes, but not always, true (so that the second
biconditional is F ↔ F and hence true), but such that there is an x making one true and the other false. For
example, we can take P (x) to mean that x is an even number (a multiple of 2) and Q(x) to mean that x is
a multiple of 3. Then an example like x = 4 or x = 9 shows that ∀x(P (x) ↔ Q(x)) is false.
Work Step by Step
We want propositional functions P and Q that are sometimes, but not always, true (so that the second
biconditional is F ↔ F and hence true), but such that there is an x making one true and the other false. For
example, we can take P (x) to mean that x is an even number (a multiple of 2) and Q(x) to mean that x is
a multiple of 3. Then an example like x = 4 or x = 9 shows that ∀x(P (x) ↔ Q(x)) is false.
