I was doing midterm review and I came across these formulas
$$\forall x \big( P(x) \to Q (x))$$
and
$$\forall x P (x) \to \forall x Q (x)$$
I wanted to know what the difference was in terms of $x$ for both of them. In the first one, if I choose $x = 5$, then it's $P(5) \to Q(5)$ but is it the same for the second one? If there are quantifiers on both, does that mean that $x$ can be a different value for both such as $P(5) \to Q(3)$?
On
The second statement says that if $P(x)$ holds of all $x$, then $Q(x)$ also holds of all $x$. Unlike the first statement, which would allow us to conclude $Q(5)$ from $P(5)$, the second statement would not allow us to conclude $Q(5)$ from $P(x)$ for any particular $x$. We would only be able to conclude $Q(5)$ if we knew $P(x)$ held for all $x$. If $P(x)$ failed to hold for even a single $x$, we would not be able to draw any conclusion about $Q(5)$. On the other hand, if $P(x)$ held for all $x$, then we could conclude that $Q(x)$ held, not just for $x=5$, but for all $x$.
Note also that the first statement entails the second statement: if $Q(x)$ holds whenever $P(x)$ holds, then if $P(x)$ holds for all $x$, we know that $Q(x)$ holds for all $x$.
In words, $\forall x\big(P(x)\to Q(x)\big)$ says that no matter what $x$ you take, if it has property $P$, then it also has property $Q$. Suppose that we’re talking strictly about integers, $P(x)$ means that $x$ is a multiple of $4$, and $Q(x)$ means that $x$ is even. Then $\forall x\big(P(x)\to Q(x)\big)$ is true: if some integer $x$ is a multiple of $4$, then $x$ is certainly even.
$\forall xP(x)\to\forall xQ(x)$, on the other hand, says that if every $x$ has property $P$, then every $x$ also has property $Q$.
These two statements are not equivalent. Suppose that the domain of discourse is the set of positive integers, $P(x)$ is the statement that $x$ is prime, and $Q(x)$ is the statement that $x$ is odd. The statement $$\forall x\big(P(x)\to Q(x)\big)$$ is false, because $2$ is prime (i.e., $P(2)$ is true), but $2$ is not odd (i.e., $Q(2)$ is false). In words, the statement says that every prime is odd, and $2$ is clearly a counterexample to that statement.
The statement $$\forall xP(x)\to\forall xQ(x)\;,$$ however, is true for the rather trivial reason that $\forall xP(x)$ is false: it’s certainly not true that all positive integers are prime! Thus, the implication is vacuously true.