Is $\forall x\exists y\in\Bbb R: (x>0\to x=y^2 +1)$ true or false and how do I determine it?

Question

It is very unclear from my textbook, how to determine whether a logical statement is true or false.

So I was wondering since the predicate logic statement is:

For all $x$ a $y$ exists in which case $x$ greater than $0$ then $x$ equals $y^2 + 1$, where the universe is all real numbers.

So lets say I choose to put $x = 2$

Then the second statement becomes:

$2 - y^2 = 1, y=1,-1$

Have I then showed that the predicate logic statement is true, because our $y$ is part of our real number universe as long as $x>0$

Thanks

Answer 1

The statements says for all $x$. So in order to prove it true, you need to show it is true for all $x \in \mathbb{R}$. However, if you can prove it false for even a single value of $x$, it is false. Hint: look for a counter example, try a very small value of $x$.