JEE Main 2024 — 27 January, Shift 1 — Question 10

Let $\mathrm{S}=\{1,2,3, \ldots, 10\}$

$\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} \cap \mathrm{B} \neq \phi ; \mathrm{A}, \mathrm{B} \in \mathrm{M}\}$

For Reflexive, M is subset of ' S ' So $\phi \in \mathrm{M}$

for $\phi \cap \phi=\phi$ $\Rightarrow$

but relation is $\mathrm{A} \cap \mathrm{B} \neq \phi$

So it is not reflexive. For symmetric, ARB $\quad \mathrm{A} \cap \mathrm{B} \neq \phi$

$\Rightarrow \mathrm{BRA} \quad \Rightarrow \mathrm{B} \cap \mathrm{A} \neq \phi$,

So it is symmetric. For transitive, If $\mathrm{A}=\{(1,2),(2,3)\}$

$$B=\{(2,3),(3,4)\}$$ $$\mathrm{C}=\{(3,4),(5,6)\}$$

ARB & BRC but A does not relate to C So it not transitive