JEE Main 2025 (Online) 2nd April Morning ShiftQuestion 1
Let $A$ be the set of all functions $f:\mathbf{Z}\to\mathbf{Z}$ and $R$ be a relation on $A$ such that $R=\{(f,g):f(0)=g(1)\text{ and }f(1)=g(0)\}$. Then $R$ is :
Sets, Relations and Function
Solution
Reflexive needs $f(0)=f(1)$ for every $f$: false. Symmetric: $fRg\Rightarrow f(0)=g(1),f(1)=g(0)\Rightarrow gRf$: true. Transitive: $fRg,gRh$ give $f(0)=g(1)=h(0)$ and $f(1)=g(0)=h(1)$, but $fRh$ needs $f(0)=h(1)$, not implied (e.g. $f(0)=h(0)=0,\ f(1)=h(1)=1$). Symmetric only: option B.