Try to give a description of $Q/ \sim$. Let $Q$ be the following subset of $\mathbb Z^2$: $$Q = \{ (a,b) \in \mathbb Z^2 \mid b \neq 0 \}$$. Define the relation $\sim$ in $Q$ by $$(a,b) \sim (c,d) \iff ad=bc$$
(i) Prove that $\sim$ is an equivalence relation in $Q$.
I have done that.
(ii) Indicate the equivalence class $[\left( 2,3\right)]$.
I have also done that.  Please see below:
$$[\left( 2,3\right)] = \{ (c,d) \in \mathbb Z^2 \mid (2,3) \sim (c,d) \} \\ $$
$\iff$
$$[\left( 2,3\right)] = \{ (c,d) \in \mathbb Z^2 \mid (2d=3c \} \\ $$
(iii) Indicate $[(a,b)]$. I have also done that.  See below
$$[(a,b)]=\{ (c,d) \in \mathbb Z^2 \mid ad=bc \}$$
(iv) Try to give a description  of $Q/ \sim$. This is my main problem. I'm not sure what I'm even asked to do. I thought the description was already presented in (iii). Maybe I should describe it geometrically ?
 A: You're supposed to recognize it. If you don't, there isn't much to be done. The $Q$ is a hint, though.
We have that $Q/\sim$ is $\Bbb Q$. This is the conventional, formal definition of the rational numbers. A pair $(a,b)\in Q$ corresponds to the fraction $\frac ab$, while $[(a,b)]$ is the collection of all fractions that represent the same rational number, through regular expanding and simplification of fractions.
A: Hint: I suspect that the problem is trying to get at the fact that this notation gives an unfamiliar look to a familiar type of object in math.  For instance, try writing out several specific elements of the equivalence class, say, $[(2,3)]$.
A: The point they're trying to get you to see is that $Q/\sim$ can be identified with the rational numbers.  That is, we have
$$
(a,b) \sim (c,d) \iff \frac{a}{b} = \frac cd
$$
so that we can effectively say that the equivalence class of $(a,b)$ is the fraction $\frac ab$.
A: Well, each equivalence class $[(a,b)]$, $b\ne 0$, can be represented as a fraction $\frac{a}{b}$ in the well-known sense.
A: Hint: provided $b$ and $d$ are non-zero, $ad = bc$ iff $a/b = c/d$, i.e., iff $a/b$ and $c/d$ represent the same rational number.
A: We have
$$\frac ab=\frac cd$$
The equivalence relation is the condition for two fractions to represent the same rational number. Thus the quotient is essentially the rational numbers. 
