Question

Let $R$ and $S$ be two non-void relations on a set $A$. Which of the following statements is false?

• A. $R$ and $S$ are transitive implies $R\cap S$ is transitive
• B. $R$ and $S$ are transitive implies $R\cup S$ is transitive
• C. $R$ and $S$ are symmetric imples $R\cup S$ is symmetric
• D. $R$ and $S$ reflexive implies $R\cap S$

Answer 1

Answer: (B)

$R$ and $S$ are transitive implies $R\cup S$ is transitive

Let $A=\{1,2,3\}$
$;R=\left\{\right(1,1),(2,2\left)\right\};S=\left\{\right(2,2),(2,3\left)\right\}$
be two transitive relations on $A$
Thus $R\cup S=\left\{\right(1,1),(1,2),(2,2),(2,3\left)\right\}$
then $(1,2)\in R\cup S$ and $(2,3)\in R\cup S$
but $(1,3)\notin R\cup S$
$R\cup S$ is not transitive