I am a bit confused regarding the set-builder notation. If I'm correct, in general, a set is described by $S = \{ x \mid p(x) \}$ where p(x) is some conditional. That's what my professor's lecture notes, say at least.
My professor's lecture notes define $\mathbb{Q} = \{\frac{p}{q} \mid p,q\in\mathbb{Z} \wedge q\neq0\}$ and define $\mathbb{N} = \{x\in \mathbb{Z} \mid x \geq 0\}$.
The way I understand the general notation, a set $S$ compromises all arbitrary objects $x$ that meet the conditional statement $p(x)$. With this understanding, the given definition of $\mathbb{Q}$ makes sense to me: it is compromised of all objects $\frac{p}{q}$ meeting some criteria. The given definition of $\mathbb{N}$, however, seems to say it is comprised of a conditional statement, $x\in\mathbb{Z}$, rather than some arbitrary object. At least, it seems to be a conditional to me based on its usage elsewhere.
Basically, is notating a set $S = \{x\in T\}$ proper? If so, why is this the case—given the qualms I have listed above? It makes much more sense to me that $S = \{x \mid x\in T\}$.
The original way of writing sets was \begin{equation} \{ x \mid \text{some property of $x$} \}. \tag{1} \end{equation} However, there is a problem, namely Russell’s paradox: The “set” $R = \{x \mid x \notin x\}$ leads to a contradiction (try to figure out whether or not $R \in R$).
The common solution to this problem is Zermelo–Fraenkel set theory. Instead of allowing arbitrary expressions like (1) above, the main way of creating new sets is to restrict the elements of an existing set. (There are a few others as well to “kickstart” the creation of sets.) Because every set now needs a parent set $P$, it makes sense to give this parent set a special place in the notation, hence \begin{equation} \{ x \in P \mid \text{some property of $x$} \}. \end{equation} This avoids the paradox above, because if you set $R = \{ x \in P \mid x \notin x \}$, you can now conclude that $R$ can’t be an element of the parent set $P$.
This explains the new notation. But what’s up with the definition of $\mathbb{Q}$ where this notation was not used? Well, in practice paradoxical sets are rare and sometimes sets are more easily defined the old way, so the previous notation is still used when it makes sense.