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Q.
If $R$ and $R'$ are symmetric relations (not disjoint) on a set A, then the relation $R\, \cap \,R'$ is
Relations and Functions - Part 2
Solution:
Given $R$ and $R'$ are not disjoint, so there is atleast one ordered pair, say, $(a$, $b) \in R \cap R'$.
$\Rightarrow (a$, $b) \in R$ and $(a$, $b)\in R'$.
As $R$ and $R'$ are symmetric relations, we get
$(b$, $a) \in R$ and $(b$, $a) \in R'$
$ \Rightarrow (b$, $a) \in R \cap R'$
Hence, $R \cap R'$ is symmetric.