Question

Prove that ∀x(P(x)∨Q(x)) and ∀xP(x)∨ ∀xQ(x) are not logically equivalent

Answer 1

Solution:

I will prove that given statements are not logically equivalent using the following example:

Domain: set of all positive numbers

P(x) = "x is an even number"

Q(x) = "x is odd number"

So, x (P(x) v Q(x)) that is (every number is either even or odd) is true.

But, x P(x) v Q(x) that is (every number is even or every number is odd) is false.

Thus, we can say that  x (P(x) v Q(x)) and x P(x) v Q(x) are not logically equivalent.

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