Answer to Question #299999 in Discrete Mathematics for shahana

Question

 Determine whether ∀x(P(x) ↔ Q(x)) and ∀x P(x) ↔ ∀xQ(x) are logically equivalent. Justify your answer.

Answer 1

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.