JEE MAIN - Mathematics (2025 - 4th April Evening Shift - No. 1)
Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$ and R be a relation on A defined by $x \mathrm{R} y$ if and only if $2 x-y \in\{0,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+\mathrm{m}+\mathrm{n}$ is equal to:
17
18
15
16
Explanation
$$\begin{aligned} &\begin{aligned} & x R y \Leftrightarrow 2 x-y \in\{0,1\} \\ & \Rightarrow \quad y=2 x \text { or } y=2 x-1 \\ & A=\{-3,-2,-1,0,1,2,3\} \\ & \mathrm{R}=\{(-1,-2),(0,0),(1,2),(-1,-3),(0,-1),(1,1), \\ & (2,3)\} \\ & \Rightarrow \quad I=7 \end{aligned}\\ &\text { For } R \text { to be reflexive }(0,0),(1,1) \in R \end{aligned}$$
But other $(a, a)$ such that $2 a-a \in\{0,1\}$
$$\Rightarrow \quad a \in\{0,1\}$$
5 other pairs needs to be added $\Rightarrow m=5$
$x R y \Rightarrow y R x$ 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, $n=5$
$\Rightarrow \quad l+m+n=17$
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