1. ## Universal quantified statements

I'm a little confused with logical equivalences of quantifications. Particularly, universal. I know that ∀x(P(x)∨Q(x)) is not logically equivalent to ∀xP(x) ∨ ∀xQ(x), but I don't know the proper way to show it.

But, what I am more concerned with is if ∀x[P(x)↔Q(x)] and ∀xP(x) ↔ ∀xq(x) are logically equivalent. I would like to understand this. Thanks

2. Originally Posted by mremwo
I'm a little confused with logical equivalences of quantifications. Particularly, universal. I know that ∀x(P(x)∨Q(x)) is not logically equivalent to ∀xP(x) ∨ ∀xQ(x), but I don't know the proper way to show it.
All humans are male or female, but...?

3. To show that ∀x(P(x)∨Q(x)) is not logically equivalent to ∀xP(x) ∨ ∀xQ(x), find an interpretation when the two formulas have different truth values. Consider the domain of natural numbers and let P(x) be "x is even" and Q(x) be "x is odd". In fact, this interpretation helps answer the second question as well.

4. I understand the first part of this problem now. What I really need help with is the latter. thanks

5. Find whether ∀x[P(x)↔Q(x)] and ∀xP(x) ↔ ∀xq(x) are true in the interpretations suggested above. This requires knowing the definition of when a formula of a certain form is true in a given interpretation (see this Wikipedia article, for example). If you are having trouble finding the truth values of these formulas, please describe what it is.