Question

Discrete mathematics counterexamples -

Supplementary question:

S1.) Give a counterexample to show that "∀x(p(x) --> Q(x)) and ∀xP(x) --> ∀xQ(x)" may not be logically equivalent.

Answer 1

Let's say we're dealing with the domain of the set of living humans. And let's define:

P(x) = x is a genetic male

Q(x) = x is a genetic female

Then ∀x P(x) says everyone is a male, which is obviously false.

And ∀x Q(x) says everyone is a female, which is also obviously false.

So the 'or' of these: ∀x P(x) ∨∀x Q(x)   

says everyone is a male or everyone is a female is false as well.

But the expression: ∀x ( P(x) ∨ Q(x) ) says:

All people are either male or female, which is true.

This one counter-example is enough to prove your expressions are not logically equivalent.