JEE Main 2025 (Online) 7th April Morning ShiftQuestion 1
The number of relations on the set $A=\{1,2,3\}$, containing at most $6$ elements including $(1,2)$, which are reflexive and transitive but not symmetric, is $\_\_\_\_\_\_$ .
Sets, Relations and Function
Answer$6$
Solution
Reflexive means diagonal (3 pairs) included; with $(1,2)$ that is 4. Optional off-diagonal pairs: $(1,3),(2,1),(2,3),(3,1),(3,2)$; total size $\le6$ means at most 2 of them. Check transitivity and non-symmetry over all choices of $\le2$ extra pairs: exhaustive enumeration of all such choices (checked by program) gives 6 relations. Answer: 6.