Ytukyg

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 $1le n,m le5, n,min mathbb{Z}$.

I have come up with a few ways to write it down but I am not sure which one (if any) is correct.

$$begin{align}
left{ frac{m}{n} vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 1\
left{ frac{m}{n} in mathbb{Q}vert n,min mathbb{Z} land1le nle5 land1le mle5right} tag 2\
left{frac{n}{m}in mathbb{Q}vert n,m in mathbb{Z}, 1le n,mle 5right} tag 3
end{align}$$

$(1)$ and $(2)$ are okay, though I would rather do it with $frac{n}{m}$ instead of $frac{m}{n}$.

In $(2)$ the part $inmathbb Q$ is redundant, but that does not harm correctness.

$(3)$ is wrong (e.g. it demands that $mleq1$)

Actually you cannot speak of "the" correct notation of..., but of "a" correct notation of...

The definition $1$ and $2$ seems correct to me, we could also use for example

$$Big{ frac{m}{n}in mathbb{Q} ,vert , n,min {1,2,3,4,5}subseteq mathbb{Z}Big} $$

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 $$forall xcolon xin cleftrightarrow x=alor x=b.$$ 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
$$forall xcolon xin cleftrightarrow xin aland phi(x).$$
We usually use the notation ${,xin Amid phi(x),}$ 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
$$forall xcolon xin cleftrightarrow exists tcolon t in aland x=F(t).$$
We usually use the notation ${,F(t)mid tin a,}$ 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:
$$tag{2'}{,qinBbb Qmid exists m,nin Bbb Zcolon (1le mle 5land 1le nle 5land q=tfrac mn),}$$

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 $exists m,ninBbb Zcolon ldots$ as a "colloquial" short form for $exists mcolon exists ncolon (min Bbb Zland ninBbb Zlandldots)$.

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 $1le n,mle 5$ 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 "$n,min Bbb Z$ and $1le n$ and $mle 5$". 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 $n,minBbb Zland 1le n,mle 5$.

You could use $$left{frac nm mid (n, m) in [[1,5]]^2right}$$ (not to sure how to obtain the notation for an integer interval on this site, but $[[1,5]]$ is a short way of writing ${1, ..., 5}$)