# The truth value of $\exists x \forall y P(x, y)$ for a given $P$.

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

• Better to read as "there exists an $x$ such that for all $y$..." – Randall Sep 25 '20 at 15:45

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 $$r=1, s=a, t=a$$.