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I need help with question as following:

  1. $X= \mathbb{Z}\times \mathbb{Z}$ I need to define the relation $R$ on $X$ as follows:

    $(X_1,X_2)R(Y_1,Y_2) \longleftrightarrow (X_1)^2+(X_2)^2=(Y_1)^2+(Y_2)^2$

Can you prove it using (X)R(Y) ?

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Since you are new, I want to give you some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. – Zev Chonoles Feb 2 '13 at 18:21
    
@akerman What parts are you having trouble with? – Git Gud Feb 2 '13 at 18:55
    
@akerman: Start with the definition of an equivalence relation. Can you state what it is you need to show in order to establish that $R$ is an equivalence relation? – JohnD Feb 2 '13 at 20:05
    
Note that $(X_1,X_2)R(Y_1,Y_2)$ if and only the corresponding points in $\mathbb{R}^2$ lie at the same distance from the origin $(0,0)$. Don't you expect this to be an equivalence relation? – 1015 Feb 3 '13 at 21:48

Hint: For describing the equivalence class, think about using cross-multiplication to determine if two fractions are equal.

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protected by Nate Eldredge Feb 3 '13 at 23:17

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