What is the correct notation for the set of rational numbers $\frac{n}{m}$ with the constraint that $1\le n,m \le5$? I have set that I want to neatly write down/present with set notation. The set contains: 
All the rational numbers $\frac{n}{m}$ with the constraint that $1\le n,m \le5, n,m\in \mathbb{Z}$.
I have come up with a few ways to write it down but I am not sure which one (if any) is correct.
$$\begin{align}
   \left\{ \frac{m}{n} \vert n,m\in \mathbb{Z} \land1\le n\le5 \land1\le m\le5\right\} \tag 1\\
   \left\{ \frac{m}{n} \in \mathbb{Q}\vert n,m\in \mathbb{Z} \land1\le n\le5 \land1\le m\le5\right\} \tag 2\\
  \left\{\frac{n}{m}\in \mathbb{Q}\vert n,m \in \mathbb{Z}, 1\le n,m\le 5\right\} \tag 3
\end{align}$$
 A: $(1)$ and $(2)$ are okay, though I would rather do it with $\frac{n}{m}$ instead of $\frac{m}{n}$.
In $(2)$ the part $\in\mathbb Q$ is redundant, but that does not harm correctness.
$(3)$ is wrong (e.g. it demands that $m\leq1$)

Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...
A: The definition $1$ and $2$ seems correct to me, we could also use for example
$$\Big\{ \frac{m}{n}\in \mathbb{Q} \,\vert \, n,m\in \{1,2,3,4,5\}\subseteq \mathbb{Z}\Big\} $$
A: in an axiomatic view of set theory, certain notations that we use are justified by corresponding axioms.
As the very simplest example of this, the notation $\{a,b\}$ for a set that has $a$ and $b$ as elements and nothing else, is justified by the Pairing Axiom which states that for any two sets $a,b$ there exists a set $c$ such that $$\forall x\colon x\in c\leftrightarrow x=a\lor x=b.$$ It also follows that this set is unique and we introduce the notiation $\{a,b\}$ as a shortcut for "the set $c$ guaranteed to exist by the pairing axiom applied to $a$ and $b$".
Likewise, the Axiom Scheme of Comprehension states that for any set $a$ and predicate $\phi$, there exists a set $c$ such that 
$$\forall x\colon x\in c\leftrightarrow x\in a\land \phi(x).$$
We usually use the notation $\{\,x\in A\mid \phi(x)\,\}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $\phi$".
Also, the Axiom Scheme of Replacement states that for every set $a$ and function $F$, there exists a set $c$ such that 
$$\forall x\colon x\in c\leftrightarrow \exists t\colon t \in a\land x=F(t).$$
We usually use the notation $\{\,F(t)\mid t\in a\,\}$ as a shortcut for "the unique set $c$ guaranteed to exist by the comprehension scheme applied to $a$ and $F$".
With this in mind, your
version $(1)$ is a formally correct notation inspired by the Axiom Schema of Replacement.
This nearly makes version $(2)$ okay as well. However, in order to make it match the Axiom Schema of Comprehension, I would - for extra strictness - prefer to see it in the following form:
$$\tag{2'}\{\,q\in\Bbb Q\mid \exists m,n\in \Bbb Z\colon (1\le m\le 5\land 1\le n\le 5\land q=\tfrac mn)\,\}$$
Note however, that in most contexts, legibility trumps strict formalism and that certain "colloqialsms" usually enter most texts. For example, in the above I myself used $\exists m,n\in\Bbb Z\colon \ldots$ as a "colloquial" short form for $\exists m\colon \exists n\colon (m\in \Bbb Z\land n\in\Bbb Z\land\ldots)$.
Finally, I suppose you mistyped something in $(3)$.
EDIT: After the original question was edited, variant $(3)$ might be understandable and might be interpreted by many to describe the intended set. However, this variant is quite ambiguous: One can interpret $1\le n,m\le 5$ as "both numbers $n,m$ are $\ge 1$ and $\le 5$". But in the same  instance, you use the comma as a logical and, thus suggesting another possible reading "$n,m\in \Bbb Z$ and $1\le n$ and $m\le 5$". I consider this ambiguity fatal. This could be saved by making it clear that you do not habitually use the comma to denote logical conjunction, i.e., write $n,m\in\Bbb Z\land 1\le n,m\le 5$.
A: You could use $$\left\{\frac nm \mid (n, m) \in [[1,5]]^2\right\}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing $\{1, ..., 5\}$)
