"The following statements are about positive real numbers. Which one is true? Explain your answer."
$\forall x, \exists y$ such that $xy < y^2$
$\exists x$ such that $\forall y, xy < y^2$
I try understanding this but the English is difficult for this problems. I think first one I say
$$xy<y^2 \iff x<y$$
so if counterexample $y=1$ is true, then $x \geq y$ is false. This statement is false.
For second statement
$$xy < y^2 \iff x<y$$
counterexample $x=1$ is true, then $y \leq x$ is false. Statement is false.
Is correct understanding? I feel doubt about my work.
As others have said in the comments, I think there's a misunderstanding of what the quantifiers mean, so the meaning of the statement is lost in translation.
$\forall x$ means "for all x," which also means "given any x, the following condition is true." It's true for all x, so if you pick any one specific x, it will be true.
$\exists x$ means "there exists an x," so it might not be true for all x, but it's true for at least one.
So let's look at the first statement: "For all $x$, there exists $y$ such that $xy<y^2$." Is it true? Can you find such a $y$, given any value of $x$?
Now, the second statement: "There exists $x$ such that, for all $y$, $xy<y^2$." Can you find an $x$ such that, no matter what $y$ you multiply it by, the result is always less than $y^2$?
You've made things harder for yourself by introducing a biconditional in both cases, and there was no reason to do so. My advice would be to either think carefully about the symbols you introduce into your translation, or do away with the symbols entirely and translate it into full sentences.