GUJCET 2025 Math Paper

Concept:A relation is symmetric if $(a,b)\in R \Rightarrow (b,a)\in R$.A relation is transitive if $(a,b),(b,c)\in R \Rightarrow (a,c)\in R$.A relation is reflexive if $(a,a)\in R$ for every $a\in A$.Explanation:Given $(1,2)\in R$, symmetry forces $(2,1)\in R$.Using transitivity on $(1,2)$ and $(2,1)$ forces $(1,1)\in R$ and $(2,2)\in R$.So every valid relation must contain $(1,1),(2,2),(1,2),(2,1)$.For the relation to be not reflexive, $(3,3)$ must be missing.If any pair involving $3$ is added, symmetry adds its reverse, and transitivity then forces $(3,3)$.So no pair with $3$ can be included.Thus the only valid relation is $R=\{(1,1),(2,2),(1,2),(2,1)\}$.Answer:The number of such relations is $1$.So the correct option is D.