1. Quantifier Proofs

Are these quantifier proofs correct? You just have to take counterexamples to disprove, and examples to prove right?

2. (i) Let $x\in\mathbf{R}$. Take $y=-x+1\Rightarrow x+y=1>0$, so the statement is true.
(ii) Let $x\in\mathbf{R}$. Take $y=x-1\Rightarrow x-y=1>0$, so the statement is true.
(iii) For $y=x-1\Rightarrow x+y=-1<0$, so the statement is false.
(iv) For $x=0\Rightarrow 0\cdot y=0,\forall y\in\mathbf{R}$, so the statement is false.
(v) If $x=0\Rightarrow 0\cdot y=0,\forall y\in\mathbf{R}$.
If $x\neq 0$, take $y=-x\Rightarrow xy=-x^2<0$, so the statement is false.
(vi) Let $x\in\mathbf{R}$. Take $y=x\Rightarrow xy=x^2\geq 0$, so the statement is true.
(vii) Take $x=0\Rightarrow 0\cdot y=0\geq 0,\forall y\in\mathbf{R}$, so the statement is true.
(viii) Take $y=-x$ and the statement is true.
(ix) $(x+y>0) \wedge (x+y=0)$ is false, so the statement is false.
(x) $(\forall x\in\mathbf{R}, \exists y\in\mathbf{R},x+y>0)$ is true (see (i)).
$(\forall x\in\mathbf{R},\exists y\in\mathbf{R},x+y=0)$ is true (take $y=-x$), so the statement is true.

3. so the way i do it (choose numbers) is wrong?

4. Originally Posted by shilz222
so the way i do it (choose numbers) is wrong?
Yes. Look at how the Dog did it it. He pick any number and should you can find a corresponding number for that one. You picked a specific number and picked a corresponing number.

5. For (3) do you mean $y = -x-1$?