Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and R be a relation on A defined by $xRy$ if and only if $2x - y \in \{0, 1\}$. 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 $l + m + n$ is equal to:
Correct Answer:
A
Explanation
$xRy \Leftrightarrow 2x - y \in \{0, 1\}$
$\Rightarrow y = 2x$ or $y = 2x - 1$
$A=\{-3,-2,-1, 0, 1, 2, 3\}$
$R = \{(-1,-2), (0,0), (1, 2), (-1, -3), (0, -1), (1, 1), (2,3)\}$
$\Rightarrow I=7$
For $R$ to be reflexive $(0,0), (1,1) \in R$
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, $n=5$
$\Rightarrow l+m+n=17$