Chapter 13: equivalence class

\[\begin{align*} & C\text{ is an equivalence class of }a\text{ on }A\\ \Leftrightarrow & \left[a\right]_{\sim}=C=\left\{ x\middle|\begin{cases} a\in A\\ x\in A\\ x\sim a\\ \sim\text{ is an equivalence relation over }A\times A=A^{2} \end{cases}\right\} \subseteq A\ne\emptyset\\ \Leftrightarrow & \left[a\right]=\left[a\right]_{\sim}=\left\{ x\middle|\begin{cases} a\in A\\ x\in A\\ x\sim a\\ \sim\text{ is an equivalence relation on }A \end{cases}\right\} \subseteq A\ne\emptyset\\ \Rightarrow & \left[a\right]_{\sim}=\left\{ x\middle|x\sim a\right\} \subseteq A\ne\emptyset \end{align*}\]

where the definition of equivalence relation can be found in 14.