Try to give a description of $Q/ \sim$.

Question

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 ?

Answer 1

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.

Answer 2

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, $[(2,3)]$.

Answer 3

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 $\frac ab$.

Answer 4

Well, each equivalence class $[(a,b)]$, $b\ne 0$, can be represented as a fraction $\frac{a}{b}$ in the well-known sense.

Answer 5

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.

Answer 6

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.