JEE MAIN - Mathematics (2024 - 31st January Morning Shift - No. 29)
Let $$A=\{1,2,3,4\}$$ and $$R=\{(1,2),(2,3),(1,4)\}$$ be a relation on $$\mathrm{A}$$. Let $$\mathrm{S}$$ be the equivalence relation on $$\mathrm{A}$$ such that $$R \subset S$$ and the number of elements in $$\mathrm{S}$$ is $$\mathrm{n}$$. Then, the minimum value of $$n$$ is __________.
Answer
16
Explanation
$$
\begin{aligned}
& A=\{1,2,3,4\} \\\\
& R=\{(1,2),(2,3),(1,4)\}
\end{aligned}
$$
$S$ is equivalence for $R < S$ and reflexive
$$ \{(1,1),(2,2),(3,3),(4,4)\} $$
for symmetric
$$ \{(2,1),(4,1),(3,2)\} $$
for transitive
$$ \{(1,3),(3,1),(4,2),(2,4)\} $$
Now set $S=\{(1,1),(2,2),(3,3),(4,4),(1,2)$, $(2, 3),(1,4),(4,3),(3,4),(2,1),(4,1),(3,2),(1,3),(3$, 1), $(4,2),(2,4)\}$
$$ n(S)=16 $$
$S$ is equivalence for $R < S$ and reflexive
$$ \{(1,1),(2,2),(3,3),(4,4)\} $$
for symmetric
$$ \{(2,1),(4,1),(3,2)\} $$
for transitive
$$ \{(1,3),(3,1),(4,2),(2,4)\} $$
Now set $S=\{(1,1),(2,2),(3,3),(4,4),(1,2)$, $(2, 3),(1,4),(4,3),(3,4),(2,1),(4,1),(3,2),(1,3),(3$, 1), $(4,2),(2,4)\}$
$$ n(S)=16 $$
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