Truth of Mathematical Statements: Which is Correct?

[HMPARTICLE]

Which one of these statements is true?

$$ \exists y >0 : \forall x > 0, y < x $$

or

$$ \forall x > 0 \exists y > 0 : y < x $$I think the second statement is correct, since for all x greater than 0, there exists at least one value of y > 0 such that y <x.

The first statement doesn't really make a lot of sense, there exists at least one value of y >0 such that for all x >0, y < x. What this says to me is that there are values of y which are less than all the values of x > 0. which can't be true since that would imply $$ y \le 0 $$

Could someone tell my why i am correct, or why i am wrong. Please!

 

[RUber]

You are right.
The real numbers should allow you to find a number smaller than any fixed positive number. So whichever one you fix, it cannot be smaller (or larger) than all others. There will always exist one smaller or larger.

 

[gopher_p]

Which one of these statements is true?

$$ \exists y >0 : \forall x > 0, y < x $$

or

$$ \forall x > 0 \exists y > 0 : y < x $$I think the second statement is correct, since for all x greater than 0, there exists at least one value of y > 0 such that y <x.

The first statement doesn't really make a lot of sense, there exists at least one value of y >0 such that for all x >0, y < x. What this says to me is that there are values of y which are less than all the values of x > 0. which can't be true since that would imply $$ y \le 0 $$

Could someone tell my why i am correct, or why i am wrong. Please!

If the point of the exercise is to understand the quantifiers and how their order matters, then I would say that your assessment is correct. To be fair, your "explanation" for why the second statement is true is really just a translation of the symbols. You should try to come up with a concrete example of a $y$ satisfying $0<y<x$.

However ...

In order to truly say whether the statements are true or false, you would first need to stipulate the universe of discourse. For instance, the second statement is true if $x$ and $y$ are real or rational variables and $<$ is the standard ordering and false if they are integer or natural number variables with the standard ordering.

The first statement can be true if you adopt a slightly nonstandard interpretation for $<$ in the right universe; e.g. take $<$ to be a reflexive order so that $\forall x:x<x$ is true.