Thread: Distinct equivalence classes of an equivalence relation on R^2

1. Distinct equivalence classes of an equivalence relation on R^2

Hi,

I'm stuck on a question for an assignment and any help would be appreciated. Here is the question:

Give a complete list of the distinct equivalence classes of the equivalence relation on R^2. For each distinct equivalence class, give a representative of the equivalence class.
(x,y)~(a,b) iff x2y = a2b

So far, I've figured out that the relation can be re-written as x2y = C, C∈ R. However, I don't how to find all the distinct equivalence classes and their representatives... I searched for guidance online but there was little information to be found on this topic.

Thanks

2. Re: Distinct equivalence classes of an equivalence relation on R^2

Equivalence classes are based on the product. For any $C \in \mathbb{R}$, we can denote the equivalence class as $[C] = \{(x,y) \in \mathbb{R}^2 | x^2y = C\}$. For a representative of the equivalence class, how about $(1,C)$? What you need to show is that:

$\displaystyle \bigcup_{C \in \mathbb{R}} [C] = \mathbb{R}^2$