JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Sets Relations and Functions

A reflexive relation on $\{1,2,3\}$ must contain $(1,1),(2,2),(3,3)$. With $(1,2)$ and $(2,3)$ present, transitivity forces $(1,3)$. The only further pairs that may be added are from $\{(2,1),(3,1),(3,2)\}$. Adding $(3,1)$ together with $(1,2)$ would force $(3,2)$, and the resulting relations either break transitivity or become symmetric. Checking the cases, the relations that remain transitive and non-symmetric are the base relation, the base together with $(2,1)$, and the base together with $(3,2)$. There are $3$ such relations.