JEE Main 2025 — 7 April, Morning Shift — Question 45

$$A=\{1,2,3\}$$

Since $R$ is reflexive,

$$(1,1),(2,2),(3,3)\in R$$

Given $(1,2)\in R$ and $R$ is not symmetric,

$$(2,1)\notin R$$

Transitivity does not force any new element initially. Possible additional pairs are

$$(1,3),(3,2),(2,3),(3,1)$$

Transitivity implies

$$(2,3)\Rightarrow(1,3), \quad (3,1)\Rightarrow(3,2)$$

Counting all valid relations with $|R|\le 6$,

$$\boxed{6}$$