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