Question

On set $A=1,2,3$, relations $R$ and $S$ are given by
$R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right)\right\}$
$S=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,3\right),\left(3,1\right)\right\}$ Then

• A. $R\cup S$ is an equivalence relation
• B. $R\cup S$ is reflexive and transitive but not symmetric
• C. $R\cup S$ is reflexive and symmetric but not transitive
• D. $R\cup S$ is symmetric and transitive but not reflexive

Answer 1

Answer: (C)

We have,
$R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\},$
$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}.$
$\therefore R\cup S=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}.$
Since, $(2,1)\in R\cup S,(1,3)\in R\cup S$
but $(2,3)\notin R\cup S$
$\therefore R\cup S$ is reflexive and symmetric but not transitive.