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I have the following statement to determine the truth value of: $$\exists x \forall y P(x, y),$$

$P(x, y)$ is the statement "$x$ divides $y$", meaning that meaning that $y = kx$ for some integer $k$. $x$ and $y$ are both positive integers.

My first issue here is my understanding of $\exists x \forall y$, which I believe translates to:

"There is some $x$ for every $y$ such that $x$ divides $y$".

Is this correct, and if so, would this result in a positive truth value as $1$ divides all values of $y$?

As an extension to this, how is best to summise my reasoning for this positive* truth value?

*or vice-versa for a false statement

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    $\begingroup$ Better to read as "there exists an $x$ such that for all $y$..." $\endgroup$ – Randall Sep 25 '20 at 15:45
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Yes, since, as you have identified, $1$ divides $a$ for all positive integers $a$,${}^\dagger$ so let $x=1$ and $y$ can be left as an arbitrary positive integer.

Also, much like as pointed out in the comments, it is better to read $\exists x\forall yP(x,y)$ as

"There exists a positive integer $x$ such that for all positive integers $y$, $x$ divides $y$."


$\dagger$: This is true by definition of division: $$r\text{ divides }s\iff \exists t\in \Bbb Z, s=tr,$$ so let $r=1, s=a, t=a$.

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