Question

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

The correct option is C $R\cup S$ is reflexive and symmetric but not transitive

Given : $A=\{1,2,3\}$

$R=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\}$ and
$S=\left\{\right(1,1),(2,2),(3,3),(1,3),(3,1\left)\right\}$

Since, $(1,1),(2,2),(3,3)\in R\cup S$ for $1,2,3\in A$

$⟹R$ is reflexive

For any $(a,b)\in R\cup S$ we get $(b,a)\in R\cup S$

$\therefore R\cup S$ is symmetric.

Now, $(2,1),(1,3)\in R\cup S$ but $(2,3)$ does not belong to $R\cup S$

$\therefore R\cup S$ is not transitive.