JEE 2025 — Mathematics PYQ

$R=\{(f,g):f(0)=g(1)\text{ and }f(1)=g(0)\}$

Reflexive: $(f,f)\in R$

$f(f)=f(0)$ and $f(1)=f(0)$ must hold $\Rightarrow$ but this is not true for all functions.

So, it is not reflexive.

Symmetric: If $(f,g)\in R\Rightarrow (g,f)\in R$

Now, $g(0)=f(1)$ and $g(1)=f(0)$ true $\therefore$ symmetric.

Transitive: If $(f,g)\in R$ and $(g,h)\in R\Rightarrow (h,f)\in R$

Now, $(f,g)\in R\Rightarrow f(0)=g(1)$ and $f(1)=g(0)$

$(g,h)\in R\Rightarrow g(0)=h(1)$ and $g(1)=h(0)$

Now, $(f,h)\in R$, we need $f(0)=h(1)$ and $f(1)=h(0)$

Hence, it is not transitive.

It is symmetric but neither reflexive nor transitive.