JEE Main 2025 — 2 April, Morning Shift — Question 29

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 $f(0) \neq f(1)$ always

Therefore, $R$ is not reflexive

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

$\because f(0)=g(1) \Rightarrow g(1)=f(0)$

and $f(1)=g(0) \Rightarrow g(0)=f(1)$

$R$ is symmetric

If $(f, g) \in R$ and $(g, h) \in R$, 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.

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