Why are the following not logically equivalent? Why are the following statements not logically equivalent? 
$$((\forall x\in D,P(x))\vee(\forall x\in D, Q(x))$$
and $$\forall x\in D, (P(x)\vee Q(x)$$
I have thought for several hours about this and I can't think of any reason why they shouldn't be equivalent. May someone help me?

For example, let $D=\mathbb R$ and $P(x)$ be $x\ge0$ and $Q(x)$ be $x<0$.  Then your first statement is not true, because both $\forall x\in D,P(x)$ and $\forall x\in D, Q(x)$ are false, but your second statement is true.
