Presenting well a sentence with quantifiers

Question

What is the syntax rule to present a syntax with quantifiers ?

Should we rather write :

\begin{equation} \forall x\in \mathbb N\quad \exists y\in\mathbb N \quad x<y\qquad (1) \end{equation}

or

\begin{equation} \forall x\in \mathbb N,\exists y\in\mathbb N,\ x<y\ ?\qquad (2) \end{equation}

Answer 1

Personally, I would write

$$\forall x\in\mathbb N\exists y\in\mathbb N : x<y$$

because it is exactly how I would read the statement:

For every integer $x$ ($\forall x\in\mathbb N$) there exists an integer $y$ ($\exists y\in\mathbb N$) such that (colon) $x$ is smaller than $y$.

but I wouldn't say the other two ways are wrong.

Answer 2

I prefer

$$\forall x \in \mathbb{N} : \exists y \in \mathbb{N} : x < y $$

but this is a matter of style, not a matter of right-or-wrong.