Question

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

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

Answer 1

The correct option is A $R$ and $S$ are transitive$\Rightarrow R\cup S$ is transitive

Let $A=\{1,2,3\}$

Let the two transitive relations on set $A$ be

$R=\{(1,1),(1,2)\}$

and $S=\{(2,2),(2,3)\}$

Now, $R\cup S=\{(1,1),(1,2),(2,2),(2,3)\}$

Hence, $(1,2),(2,3)\in R\cup S$

$\Rightarrow (1,3)\in R\cup S$

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