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What are equivalence classes in set theory?
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Definition: Let $\sim$ be an equivalence relation on a set $\mathrm{A}$ and let $a \in A$. Then the equivalence class of a is defined to be:

$\mathrm{cl}(\mathrm{a})=\{b \in A \mid \mathrm{a} \sim \mathrm{b}\}$

Example:

If $a \sim b$ with $2 \mid(a-b)$, what is the equivalence class of zero?
Well, $\mathrm{cl}(0)=\{b \in A \mid 0 \sim \mathrm{b}\}=c l(a)=\{b \in A|2|(0-\mathrm{b})\}=c l(a)=\{b \in A|2| \mathrm{b}\}=2 \mathbb{Z}$ (the set of all even numbers.) So $\mathrm{cl}(0)=2 \mathbb{Z} .$
What then is $\mathrm{cl}(1) ?$ Going through the same steps, $\mathrm{cl}(1)=2 \mathbb{Z}+1$

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