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 $\Rightarrow R \cup S$ 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 $\Rightarrow R \cup S$ is reflexive

Answer 1

Answer: (A)

Let $A = \{1$, $2$, $3\}$, and let $R = \{(1$, $1)$, $(1$, $2)\}$,
$S = \{(2$, $2)$, $(2$, $3)\}$ be transitive relation on $A$.
Then, $R \cup S = \{(1$, $1)$, $(1$, $2)$, $(2$, $2)$, $(2$, $3)\}$.
$R \cup S$ is not transitive, since $(1$, $2) \in R \cup S$ and
$(2$, $3) \in R \cup S$ but $(1$, $3) \notin R \cup S$.