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I'm really confused about the equivalence class of the following relation:

On $\,\Bbb R^2,\;xRy\;$ if $\;|x|=|y|\;$ where $\,|x|=\sqrt{x_1^2+x_2^2}\,.$

Angelo
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Denise Sullivan
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    Rephrased... two points are equivalent "if they are on the same circle centered at the origin" or alternately phrased "if they are the same distance from the origin." The equivalence classes themselves? The circles centered at the origin. – JMoravitz May 30 '23 at 18:34
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – CrSb0001 May 30 '23 at 18:39
  • @JMoravitz Those circles, and also the singleton set ${(0,0)}$. – aschepler May 30 '23 at 18:47

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Two points are related if they are if their distance to the origin is the same. That's clearly an equivalence relationship and the classes are the collection of circles centered at the origin with radius $r>0$ and the point $(0, 0) $, that we can consider as a circle of radius $r=0$, so $$\mathcal{R}=\{\{x^2+y^2=r^2\}:r\geq 0\}$$

Guillermo García Sáez
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