Can someone help me with understanding what this statement is actually saying?

Question

$$\forall x\in \mathbb R\,\exists y\in \mathbb R\text{ such that }x+y>0\implies xy>0$$

Can someone explain to me Why this statement is True and how the Quantifiers effect it?

Answer 1

With quantified statements you can think of it as a game:

Someone gives you an $x \in \mathbb R$, then you have to find an $y \in \mathbb R$ such that the implication $$x+y>0 \implies xy>0$$ is true.

Now you can see, that this statement is always true. Because for every $x \in \mathbb R$ you're given, you can find an $y \in \mathbb R$ such that the implication is always true. Just define your $y \in \mathbb R$ as $$y=-x$$ and the expression $x+y>0$ will always be false, which makes the implication always true. (Anything can be deduced from something that is false.)

Hope that helps!