JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Sets Relations and Functions

$R=\{(1,2),(2,3),(1,4)\}$ on $A=\{1,2,3,4\}$. Reflexive and symmetric closure adds $(1,1),(2,2),(3,3),(4,4),(2,1),(3,2),(4,1)$.

Transitivity: $(1,2)$ and $(2,3)$ force $(1,3)$ and, by symmetry, $(3,1)$. Then $(2,1)$ and $(1,4)$ force $(2,4)$ and $(4,2)$. Continuing, since $1$ is now related to $2$, $3$ and $4$ (and vice versa by symmetry), transitivity forces every pair among $\{1,2,3,4\}$ to be related — the four elements collapse into a single equivalence class.

So $S=A\times A$, with $n=16$ elements. This is already the smallest such equivalence relation, since $R$ itself already chains all four elements together, forcing the whole class to merge.

Minimum value of $n$ is $16$.