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
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.
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.