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1.Relation and Function
normal
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 ==> $R \cup S$ is transitive
B
$R$ and $S$ are transitive ==> $R \cap S$ is transitive
C
$R$ and $S$ are symmetric ==> $R \cup S$ is symmetric
D
$R$ and $S$ are reflexive ==> $R \cap S$ is reflexive
Solution
(a) Let $A = \{ 1,\,2,\,3\} $ and $R = \{(1, 1), (1, 2)\}, S = \{(2, 2) (2, 3)\}$ be transitive relations on $A$.
Then $R \cup S = \{(1, 1); (1, 2); (2, 2); (2, 3)\}$
Obviously, $R \cup S$ is not transitive. Since $(1, 2)$ $ \in $ $R \cup S$ and $(2,\,3) \in R \cup S$ but $(1, 3)$ $ \notin R \cup S$.
Standard 12
Mathematics
