JEE Main 2025 — Sets And Relations Question with Solution

$$\begin{array}{rl}& \begin{array}{rl}& xRy\iff 2x-y\in \{0,1\}\\ & \Rightarrow y=2x\text{ or }y=2x-1\\ & A=\{-3,-2,-1,0,1,2,3\}\\ & \mathrm{R}=\{(-1,-2),(0,0),(1,2),(-1,-3),(0,-1),(1,1),\\ & (2,3)\}\\ & \Rightarrow I=7\end{array}\\ & \text{ For }R\text{ to be reflexive }(0,0),(1,1)\in R\end{array}$$

But other $(a,a)$ such that $2a-a\in \{0,1\}$

$$\Rightarrow a\in \{0,1\}$$

5 other pairs needs to be added $\Rightarrow m=5$

$xRy\Rightarrow yRx$ to be symmetric

$(-1,-2)\Rightarrow (-2,-1)$

$(1,2)\Rightarrow (2,1)$

$(-1,-3)\Rightarrow (-3,-1)$

$(0,-1)\Rightarrow (-1,0)$

$(2,3)\Rightarrow (3,2)\Rightarrow 5$ needs to be added, $n=5$

$\Rightarrow l+m+n=17$