Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.
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suppose domain consist of all positive integers
p(x) =x is divisible by 2 and q(x) =x is divisible by 4
we know that it is not always true that all positive integers are divisible by 2.
consider ∀xP(x) → ∀xQ(x) ,we know that ∀xP(x) is always false so this proposition must be true.
now consider ∀x(P(x) → Q(x)) this proposition is false always because it is saying that "if a number x is divisible by 2, implies that it is divisible by 4".
Hence, we conclude that they are not logically equivalent.