JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Sets Relations and Functions

Start with the required reflexive pairs on $\{1,2,3,4\}$: $(1,1),(2,2),(3,3),(4,4)$ — 4 pairs.

$R$ must contain $(1,2)$ and $(1,3)$; symmetry forces $(2,1)$ and $(3,1)$ as well.

Transitivity: $(2,1)$ and $(1,3)$ force $(2,3)$; $(3,1)$ and $(1,2)$ force $(3,2)$.

So the minimum set of pairs is
$(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)$ — a total of $10$ elements, and this set is already reflexive, symmetric and transitive.