Question:
Let $A=(1,2,3,4)$ and $R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(3,2)\} .$ Show that $R$ is reflexive and transitive but not symmetric.
Solution:
Given that, $A=\{1,2,3\}$ and $R=\{1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(3,2)\}$
Now,
$\mathrm{R}$ is reflexive $\because(1,1),(2,2),(3,3),(4,4) \in \mathrm{R}$
$\mathrm{R}$ is not symmetric $\because(1,2),(1,3),(3,2) \in \mathrm{R}$ but $(2,1),(3,1),(2,3) \notin \mathrm{R}$
$\mathrm{R}$ is transitive $\because(1,3) \in \mathrm{R}$ and $(3,2) \in \mathrm{R} \Rightarrow(1,2) \in \mathrm{R}$
Thus, $R$ is reflexive and transitive but not symmetric.