The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

<p>An equivalence relation on a finite set is uniquely determined by its partition into equivalence classes. Hence, counting the number of equivalence relations on a set is equivalent to counting the number of ways to partition that set.</p> <hr /> <h3>Step: Counting partitions of $\{1,2,3\}$</h3> <p>We want all possible ways to split the set $\{1,2,3\}$ into nonempty subsets (its “blocks”).</p> <p><p><strong>3 blocks (each element in its own block)</strong> </p> <p>$ \{\{1\}, \{2\}, \{3\}\}. $</p></p> <p><p><strong>2 blocks</strong> </p></p> <p><p>$\{\{1,2\}, \{3\}\}$ </p></p> <p><p>$\{\{1,3\}, \{2\}\}$ </p></p> <p><p>$\{\{2,3\}, \{1\}\}$</p></p> <p><p><strong>1 block</strong> (all elements together) </p> <p>$ \{\{1,2,3\}\}. $</p></p> <p>Counting these, there are a total of <strong>5</strong> distinct partitions, and thus <strong>5</strong> equivalence relations on the set $\{1,2,3\}$. </p> <p>All equivalence relations are automatically nonempty (they include at least $(1,1), (2,2), (3,3)$ because they are reflexive), so the answer to “the number of nonempty equivalence relations” is also <strong>5</strong>.</p> <hr /> <h4><strong>Answer: Option C (5)</strong></h4>