Understanding set notation and which well known set is this?

Question

First of all I wan't to ask if my understanding of the notation is correct, and later on, what well known set is this one?

I'm not allowed to embed images yet, so here is the link to the set:

For elements (n) of natural numbers set (N) such that (elements are bigger than one) and (for all elements x,y in natural numbers set), not sure how to transition in to the brackets, but I understand it as follows, [(elements x,y multiplied together are equal to the elements we are looking for initialy (n), this implies (=>) that always either x or y are equal to 1 in every case belonging to the set we are looking for)].

Also there is an appendix that (IN THIS COURSE MATERIAL) the natural set begins from 1 and not from 0, to get rid of all the confusions that this may imply. Thanks.

Answer 1

Let's read it from left to right first.

$\{ n \in \mathbb{N} \mid \cdots$

The set of natural numbers $n$ such that ...

$\cdots (n > 1) \wedge \cdots$

... $n$ is greater than $1$ and ...

$\cdots (\forall x,y \in \mathbb{N})[ \dots$

... for all natural numbers $x,y$ ...

$\cdots (xy=n) \Rightarrow \cdots$

... if $xy=n$, then ...

$\cdots (x=1 \vee y=1)]\}$

... $x=1$ or $y=1$.

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So it's the set of natural numbers $n$ which are greater than $1$ and satisfy the property that if you write them as a product of natural numbers, then one or the other of those natural numbers must be equal to $1$. But these are exactly the prime numbers!