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 (integer)
16
Solution
$$
\begin{aligned}
& A=\{1,2,3,4\} \\\\
& R=\{(1,2),(2,3),(1,4)\}
\end{aligned}
$$
<br/><br/>$S$ is equivalence
for $R < S$ and reflexive
<br/><br/>$\{(1,1),(2,2),(3,3),(4,4)\}$
<br/><br/>for symmetric
<br/><br/>$\{(2,1),(4,1),(3,2)\}$
<br/><br/>for transitive
<br/><br/>$\{(1,3),(3,1),(4,2),(2,4)\}$
<br/><br/>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), $(4,2),(2,4)\}$
<br/><br/>$n(S)=16$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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