Question

The relation $R$ and $R^{\prime }$ are symmetric in the set $A$, then show that $R\cup R^{\prime }$ and $R\cap R^{\prime }$ are symmetric.

Answer 1

We will prove, union of sets is symmetric
Let $S=R\cup R^{\prime }$
Let $(a,b)\in S$
$\Rightarrow (a,b)\in R\cup R^{\prime }$
$\Rightarrow (a,b)\in R$ or $(a,b)\in R^{\prime }$
$\Rightarrow (b,a)\in R$or $(b,a)\in R^{\prime }$ (Since, R and R' are symmetric)
$\Rightarrow (b,a)\in R\cup R^{\prime }$
$\Rightarrow (b,a)\in S$
Hence, $S=R\cup R^{\prime }$ is symmetric.
Now, we will prove, intersection of sets is symmetric

Let $P=R\cap R^{\prime }$
Let $(a,b)\in P$
$\Rightarrow (a,b)\in R\cap R^{\prime }$
$\Rightarrow (a,b)\in R$ and $(a,b)\in R^{\prime }$
$\Rightarrow (b,a)\in R$ and $(b,a)\in R^{\prime }$ (Since, R and R' are symmetric)
$\Rightarrow (b,a)\in R\cap R^{\prime }$
$\Rightarrow (b,a)\in P$
Hence, $P=R\cap R^{\prime }$ is symmetric.