= Solution
An <equivalence relation> is a <binary relation> that is simultaneously a <reflexive relation>, a <symmetric relation> and a <transitive relation>: $x\sim x$; $x\sim y$ implies $y\sim x$; and $x\sim y$, $y\sim z$ imply $x\sim z$. The <equivalence class> of $x$ is
$$
[x]=\{y\in X:y\sim x\}.
$$
Each <equivalence class> is nonempty by <reflexivity>, and every $x$ belongs to $[x]$, so the classes cover $X$. If $z\in[x]\cap[y]$, then the <symmetric relation> and <transitive relation> properties give $x\sim z\sim y$. Every $w\in[x]$ therefore satisfies $w\sim x\sim y$, so $[x]\subseteq[y]$; the reverse inclusion follows similarly. Thus two classes are either identical or disjoint. The distinct <equivalence classes> consequently form a <set partition> of $X$.
The divisibility-comparability relation is a <reflexive relation> and a <symmetric relation>, but \b[is not an equivalence relation]. Indeed, $2\sim6$ and $6\sim3$, whereas neither $2\mid3$ nor $3\mid2$, so <transitivity> fails.
For the requested four-class construction, partition the positive integers into
$$
C_1=\{1\},\quad C_2=\{2\},\quad
C_3=\{3,5,7,\ldots\},\quad C_4=\{4,6,8,\ldots\}.
$$
Define $x\approx y$ exactly when they lie in the same $C_j$. Equality of the class label is an <equivalence relation>. \b[Its four classes are the two finite singletons and the two infinite parity classes displayed above.]
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