= Solution
For an <equivalence relation> $\sim$ on $X$, the <equivalence class> of $x\in X$ is $[x]=\{y\in X:y\sim x\}$. A <set partition> is a family of nonempty, pairwise disjoint subsets whose union is $X$.
Reflexivity ensures $x\in[x]$, so every <equivalence class> is nonempty and every element belongs to one. If $[x]\cap[y]$ contains $z$, then $z\sim x$ and $z\sim y$; symmetry and transitivity give $x\sim y$. If $w\in[x]$, transitivity gives $w\sim y$, so $w\in[y]$; interchanging $x,y$ gives the reverse inclusion. Thus intersecting <equivalence classes> are equal.
Taking the collection of distinct <equivalence classes> therefore produces a <set partition> of $X$. \b[Every element lies in exactly one class.] If $X$ is empty, the empty family is its <set partition>.
Back to article page
