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What is the equivalence class of 0?

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  • $\begingroup$ You haven't proved that your relation is symmetric. On the other hand, a~0 iff $2\mid a+0$. Can you conclude from that? $\endgroup$ – Xam Dec 12 '16 at 1:27
  • $\begingroup$ Let $a\in\mathbb{Z}$. Clearly, $2|a+a$. Thus, $a\sim a$. Hence, reflexive property holds. $\endgroup$ – Juniven Dec 12 '16 at 1:31
  • $\begingroup$ @ash Please do not deface your question, in courtesy to those who already commented or answered, and also to others who may find it useful in the future. $\endgroup$ – dxiv Dec 12 '16 at 2:32
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Hint:

This relation is nothing else than a and b have the same parity.

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The equivalence class of $0$ is given by \begin{align} [0]&=\{a\in\mathbb{Z}:a\sim 0\}\\ &=\{a\in\mathbb{Z}:2|a\}\\ &=\{\dots, -4, -2, 0, 2, 4, \dots\} \end{align}

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