Definition:Equivalence Class

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Definition

Let $\mathcal R$ be an equivalence relation on $S$

Let $x \in S$.


Then the equivalence class of $x$ under $\mathcal R$, or the $\mathcal R$-equivalence class of $x$, or just the $\mathcal R$-class of $x$, is the set:

$\left[\!\left[{x}\right]\!\right]_\mathcal R = \left\{{y \in S: \left({x, y}\right) \in \mathcal R}\right\}$


Thus:

$y \in \left[\!\left[{x}\right]\!\right]_\mathcal R \iff \left({x, y}\right) \in \mathcal R$


If $\mathcal R$ is an equivalence on $S$, then each $t \in S$ that satisfies $\left({x, t}\right) \in \mathcal R$ (or $\left({t, x}\right) \in \mathcal R$) is called a $\mathcal R$-relative of $x$.


That is, the equivalence class of $x$ under $\mathcal R$ is the set of all $\mathcal R$-relatives of $x$.

This construction is justified by Relation Partitions a Set iff Equivalence.


Notation

The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif.


Other variants:


Also see


Sources

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