= Solution
The <equivalence class> of $x\in X$ is $[x]=\{y\in X:y\sim x\}$. By the reflexivity of the <equivalence relation>, $x\in[x]$, so every <equivalence class> is nonempty and their union is $X$.
Suppose $[x]\cap[z]\ne\varnothing$, and choose $w$ in this <intersection>. We have $w\sim x$ and $w\sim z$. Symmetry and transitivity give $x\sim z$. If $y\in[x]$, then $y\sim x\sim z$, so $y\in[z]$; interchanging $x,z$ proves the reverse inclusion. Thus two <equivalence classes> are either identical or disjoint. \b[The distinct <equivalence classes> therefore form a <set partition> of $X$.] This also covers $X=\varnothing$, whose family of <equivalence classes> is empty.
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