Mathematics · Sets and Relations

JEE Main 2025 — 4 April, Evening Shift — Question 27

Let $A=\{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $x R y$ if and only if

$2 x-y \in\{0,1\}$ . Let $I$ 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 $I+m+n$ is equal to:

Answer: C

$x R y \Leftrightarrow 2 x-y \in\{0,1\}$

$\Rightarrow y=2 x$ or $y=2 x-1$

$A=\{-3,-2,-1,0,1,2,3\}$

$R=\{(-1,-2),(0,0),(1,2),(-1,-3),(0,-1),(1,1)$ , $(2,3)\}$

$\Rightarrow \quad I=7$

For $R$ to be reflexive $(0,0),(1,1) \in R$

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, $n=5$

$\Rightarrow I+m+n=17$

Answer key and solution verified before publishing.

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