JEE MAIN - Mathematics (2023 - 13th April Evening Shift - No. 16)
Let $$\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$$ and $$\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$$ or $$\left.b^{2}=a+1\right\}$$ be a relation on $$\mathrm{A}$$. Then the minimum number of elements, that must be added to the relation $$\mathrm{R}$$ so that it becomes reflexive and symmetric, is __________
Answer
7
Explanation
$$
\begin{aligned}
A & =\{-4,-3,-2,0,1,3,4\} \\\\
R= & \{(-4,4),(-3,3),(0,0),(1,1) \\
& (3,3),(4,4),(0,1),(3,-2)\}
\end{aligned}
$$
Relation to be reflexive $(a, a) \in R \forall a \in A$
$\Rightarrow (-4,-4),(-3,-3),(-2,-2)$ also should be added in $R$.
Relation to be symmetric if $(a, b) \in R$, then $(b, a) \in R \forall a, b \in A$
$\Rightarrow (4,-4),(3,-3),(1,0),(-2,3)$ also should be added in $R$
$\Rightarrow$ Minimum number of elements to be added to $R=3+4=7$
Relation to be reflexive $(a, a) \in R \forall a \in A$
$\Rightarrow (-4,-4),(-3,-3),(-2,-2)$ also should be added in $R$.
Relation to be symmetric if $(a, b) \in R$, then $(b, a) \in R \forall a, b \in A$
$\Rightarrow (4,-4),(3,-3),(1,0),(-2,3)$ also should be added in $R$
$\Rightarrow$ Minimum number of elements to be added to $R=3+4=7$
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