JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Sets Relations and Functions

An equivalence relation on a finite set corresponds exactly to a partition of that set into disjoint non-empty blocks, with two elements related precisely when they lie in the same block. So counting equivalence relations on $\{1,2,3\}$ reduces to counting partitions of a 3-element set.

The partitions of $\{1,2,3\}$ are:
$\{1\},\{2\},\{3\}$ — all singleton
$\{1,2\},\{3\}$
$\{1,3\},\{2\}$
$\{2,3\},\{1\}$
$\{1,2,3\}$ — one block

That is $1+3+1=5$ partitions, hence 5 equivalence relations. Every one of them contains the diagonal pairs $(1,1),(2,2),(3,3)$, so none is the empty relation — all 5 qualify as "non-empty."

The number of non-empty equivalence relations on $\{1,2,3\}$ is 5.