Question

On set $A=\{1,2,3\}$, relations $R$ 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

$R\cup S=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right(1,3),(3,1\left)\right\}$
$R\cup S$ is reflexive because all $(1,1),(2,2),(3,3)$ are present.

$R\cup S$ is symmetric because $(b,a)\in R\cup S$ for each $(a,b)\in R\cup S$

It is not transitive because $(3,1),(1,2)$ is present in $R\cup S$ but $(3,2)$ is not present in $R\cup S$.