How to work with a statement: "for all $x$, if $p(x)$ then $q(x)$" (contradiction and contraposition)

Question

Proving by contradiction and contrapositive a statement of the sort "for all $x$, if $p(x)$ then $q(x)$"

Question about the notation: is it equivalent to $\forall x (p(x)\to q(x))$ or $(\forall x (p(x))\to q(x))$?

I know how to prove by contradiction and contra position the latter statement.

I want to make sure I know what to do with other statement: $\forall x (p(x)\to q(x))$

Proof by contradiction is: $\exists x (p(x)\wedge \neg q(x))$

Proof by contra position: $\forall x (p(x)\to q(x))\iff (\forall x (p(x))\to \forall x (q(x)))\iff (\exists x (q(x))\to \exists x (p(x)))$

Are these correct?

Answer 1

The statement "for all $x$, if $p(x)$ then $q(x)$" is equivalent to $\forall x:(p(x) \rightarrow q(x))$

The statement $(\forall x:p(x)) \rightarrow q(x)$ is equivalent to "If for all $x$ $p(x)$, then $q(x)$."

The statement $\forall x (p(x) \rightarrow q(x))$ is not equivalent to $\forall x p(x) \rightarrow \forall x q(x)$. For example the sentence "For every car: if it is not moving then it is still" is equivalent to "For every car: it is moving or it is still" but this is NOT the same as "For every car:it is moving OR For every car: it is still".

Answer 2

$$ \forall x\big(p(x)\big) \,\to\, q(x) $$ cannot be correct, since the first part, the assumption, $\forall x\big(p(x)\big)$ DOES NOT depend on $x$, while the second part, the conclusion, $q(x)$ DOES depend.