Subsection 8.5.1 (01TA): Equivalence Classes

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Table of Contents
Part 2: Relations
Chapter 8: Equivalence Relations and Apartness Relations
Section 8.5: Quotients by Equivalence Relations
Subsection 8.5.1: Equivalence Classes (cite)

8.5.1 Equivalence Classes

Let $A$ be a set, let $R$ be a relation on $A$, and let $a\in A$.

Definition 8.5.1.1.1 ▶ Equivalence Classes

The equivalence class associated to $a$ is the set $\webleft [a\webright ]$ defined by

\begin{align*} \webleft [a\webright ] & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ x\sim _{R}a\webright\} \\ & = \webleft\{ x\in X\ \middle |\ a\sim _{R}x\webright\} \tag {since $R$ is symmetric}.\end{align*}

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