Sets, Relations & Functions
Counting Relations with Given Properties
nta_pyq_2025_apr
Grade 11
Let $A = \{1, 2, 3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ___
Step-by-Step Solution
Key Concept: Any reflexive transitive relation containing $(1,2)$ and $(2,3)$ must contain $(1,1),(2,2),(3,3),(1,3)$. Then check which additional pairs can be included while maintaining transitivity but breaking symmetry.
Minimum relation $R_1 = \{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$ is reflexive, transitive, not symmetric. Adding $(3,2)$ gives $R_2$ (still valid). Adding $(2,1)$ gives $R_3$ (still valid). These 3 relations work. Total $= 3$.
Correct Answer: 3