Question

If $R_1$ and $R_2$ be symmetric relations in a set $A$, then $R_1 \cup R_2$ is

• A. reflexive
• B. transitive
• C. symmetric
• D. None of these

Answer 1

Answer: (C)

As $R_1$ and $R_2$ are relations in $A$
$\therefore R_{1 } \subseteq A \times A$ and $R_{2} \subseteq A \times A$
$\therefore R_{1} \cup R_{2} \subseteq A \times A, so R_{1} \cup R_{2}$ is a relation in $A$.
Let $(a, b) \in R_1 \cup R_2$
$\Rightarrow (a, b) \in R_1$ or $(a, b) \in R_2$
$\Rightarrow (b, a) \in R_1$, or $(b, a) \in R_2$
$(\because R_1, R_2$ are symmetric)
$\Rightarrow (b, a) \in R_1 \cup R_2$
Thus $(a, b) \in R_1 \cup R_2$
$\Rightarrow (b, a) \in R_1 \cup R_2$
$\Rightarrow R_1 \cup R_2$ is symmetric