Truth value of quantified statement

Question

I have $P(x,y) \iff xy=1$; the universe of discourse for $x$ is the set of positive integers, and the universe of discourse for $y$ is the set of real numbers. $$ \forall x \exists y P(x,y) $$

I'm confused about reading the above statement. I'm quoting above statement from a book that says the statement read as "For every positive integer $x$ there is a real number $y$" such that $xy=1$. so the statement is true...I want to know what value, that makes this statement true, of $y$. is there only one value of $y$ for all values of $x$? or I need to change the value of $y$ every time when i change $x$.

Please If someone could explain it would be appreciated. thank you.

Answer 1

$\forall x \in \mathbb Z^+, \exists y \in \mathbb R$ such that $xy = 1.$ True. The value of y that makes the statement true will change according to the value of $x$: $(x, y): (2, \frac12), (8, \frac 18), ...$. we can take any integer $x$, and put $y = \frac 1x$ so that $xy = 1$ is true. In truth, we can say that every positive integer has a multiplicative inverse, y, in $\mathbb Q \subset \mathbb R$

The expression $\exists y \forall x, P(x, y)$ is false when $P(x, y)$ means $xy = 1$. That is, there does not exist a value of $y \in \mathbb R,$ such that for each and every positive integer , we have that $xy=1$.