Let A=1,2,3,4 and R=(1,2),(2,3),(1,4) be a relation on A.

Question

Let $A={1,2,3,4}$ and $R={(1,2),(2,3),(1,4)}$ be a relation on $A$. Let $S$ be the equivalence relation on $A$ such that $R\subset S$ and the number of elements in $S$ is $n$. Then, the minimum value of $n$ is \_\_\_\_\_\_\_\_\_\_.

Answer: 16

Accepted Answer

$\begin{gathered} A={1,2,3,4} \ R={(1,2),(2,3),(1,4)} \end{gathered}$

$S$ 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), $(4,2),(2,4)}$

$n(S)=16$

Solutions · 1