Question

Let $A=\{1,2,3,4\}$ and $R$ be a relation in $A$ given by $R=\left\{\right(1,1),(2,2\left)\right(3,3),(4,4),(1,2),(3,1),(1,3\left)\right\}$ then $R$ is :

• A. Reflexive and transitive only
• B. Transitive and symmetric only
• C. An equivalence relation
• D. None of the above

Answer 1

Answer: (A)

Reflexive and transitive only

$A=\{1,2,3,4\}$
$R=\left\{\right(1,1),(2,2),(3,3),(4,4),(1,2),(3,1\left)\right(1,3\left)\right\}$
clearly $(1,1)ϵR,(1,2)ϵ(2,2)ϵR,(3,3)ϵR,(4,4)ϵR$
$\Rightarrow R$ is Reflective.
also $(1,1)ϵR,(1,2)ϵR\Rightarrow (1,2)R$
and $(1,2)ϵR,(1,3)ϵR\Rightarrow (1,3)ϵR\Rightarrow R$ is transitive
But for $(1,2)ϵR(2,1)\notin R,\Rightarrow R$ is not symmetric
$\Rightarrow R$ is reflective and transitive only.