Question

If $R$ and $R^{\prime }$ are symmetric relations (not disjoint) on a set A, then the relation $R\cap R^{\prime }$ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

Answer: (B)

Given $R$ and $R^{\prime }$ are not disjoint, so there is atleast one ordered pair, say, $(a$, $b)\in R\cap R^{\prime }$.
$\Rightarrow (a$, $b)\in R$ and $(a$, $b)\in R^{\prime }$.
As $R$ and $R^{\prime }$ are symmetric relations, we get
$(b$, $a)\in R$ and $(b$, $a)\in R^{\prime }$
$\Rightarrow (b$, $a)\in R\cap R^{\prime }$
Hence, $R\cap R^{\prime }$ is symmetric.