Q. 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 implies $R \cup S$ is symmetric.
D) $R$ and $S$ are reflexive implies $R \cap S$ is reflexive.
Explanation
(b)
Let $A = \{ 1,\ 2,\ 3\}$
Let two transitive relations on the set $A$ are
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R = \{(1,\ 1),\ (1,\ 2)\}$
And $\ \ \ \ \ \ \ \ \ \ S = \{(2,\ 2),\ (2,\ 3)\}$
Now, $R \cup S = \{(1,\ 1),\ (1,\ 2),\ (2,\ 2),\ (2,\ 3)\}$
Here, $(1,\ 2),\ (2,\ 3) \in R \cup S \Rightarrow (1,\ 3) \notin R \cup S$
$\therefore R \cup S$ is not transitive.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.