JEE Main 2025 — Sets And Relations Question with Solution

For $R$ to be reflexive, $(f,f)$ must be in $R$.

The means $f(0)=f(1)$ and $f(1)=f(0)$ must be true for all $f$.

But $f(0)\ne f(1)$ always

Therefore, $R$ is not reflexive

If $(f,g)\in R$, then $f(0)=g(1)$ and $f(1)=g(0)$

$$\begin{array}{rl}& \because f(0)=g(1)\Rightarrow g(1)=f(0)\\ & \text{ and }f(1)=g(0)\Rightarrow g(0)=f(1)\end{array}$$

$R$ is symmetric

If $(f,g)\in R$ and $(g,h)\in \mathrm{R}$, then $f(0)=g(1)$, $f(1)=g(0),g(0)=n(1)\&g(1)=h(0)$

Since, $f(0)=g(1)$ and $g(1)=h(0)$, then $f(0)$ is not necessarily equal to $h(0)$.

Therefore, $R$ is not transitive.

$\therefore$ The relation $R$ is symmetric but not reflexive or transitive.