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How can I describe properly the set of all equivalence classes?

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  • $\begingroup$ Are you familiar with polar coordinates? $\endgroup$ – Joe Aug 30 '19 at 15:09
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You can write the relationship as $z_1Rz_2\iff|z_1|=|z_2|$. Is now very easy to understand that $\mathbb{C}/R$ is the set of all origin centered circumferences of the complex plane plus the origin. Each one is the equivalence class of any of it's points.

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As other have noted, each circle centred on $0$ is an equivalence class, since $R$ is equivalent to modulus equality. However, there is a subtlety: while circles are uncountable equivalence classes, there's another equivalence class, $\{0\}$, with only one element.

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