JEE 2024 — Mathematics PYQ

Given $S=\{1,2,3,\dots ,10\}$

Also, $M=P(S)$ = power set of $S$

$(i)$ Let $A,B\in M$

$R=\{(A,B):A\cap B=\varnothing ,A,B\in M\}$  \hfill [Given]

If $A=B\Rightarrow A\cap B\ne \varnothing$

Hence, $R$ is not reflexive.

$(ii)$ As if $(A,B)\in M\Rightarrow (B,A)\in M$

So, $R$ is symmetric.

$(iii)$ Let $(A,B)\in M$ and $(B,C)\in M$

$\Rightarrow A\cap B=\varnothing$ and $B\cap C=\varnothing$

But $A\cap C$ not necessarily empty

So, $(A,C)\notin M$

So, $R$ is not transitive.