= Solution
A <binary relation> $R$ on $A$ is an <equivalence relation> exactly when it is <reflexive>, <symmetric>, and <transitive>:
$$
\begin{gathered}
aRa\quad(a\in A),\\
aRb\Longrightarrow bRa\quad(a,b\in A),\\
aRb\text{ and }bRc\Longrightarrow aRc\quad(a,b,c\in A).
\end{gathered}
$$
The <equivalence class> of $a\in A$ is \b[$[a]_R=\{b\in A:aRb\}$]. The <equivalence classes> partition $A$: every element belongs to its own class, and two classes are either equal or disjoint.
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