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

  1. #1
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    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
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  2. #2
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    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:

    $\displaystyle \bigcup_{C \in \mathbb{R}} [C] = \mathbb{R}^2$
    Last edited by SlipEternal; Oct 4th 2017 at 08:42 AM.
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