Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ______ -

Question

Aicharya · Accepted Answer

$R$ is reflexive $\Rightarrow R$ have $(1,1),(2,2),(3,3)$
$R$ is transitive
$\begin{aligned}
& \because(1,2),(2,3) \in R \quad \therefore(1,3) \in R \\ & \therefore \quad R_1=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}
\end{aligned}$
Clearly $R_1$ is reflexive and transitive but not symmetric.
Similarly,
$\begin{aligned}
& R_2=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3),(3,2)\} \\ & R_3=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3),(2,1)\}
\end{aligned}$
Therefore, 3 relations are possible

Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and…

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