Question:
If $R$ is a symmetric relation on a set $A$, then write a relation between $R$ and $R^{-1}$
Solution:
Here, R is symmetric on the set A.
Let $(a, b) \in R$
$\Rightarrow(b, a) \in R$ [Since $R$ is symmetric]
$\Rightarrow(a, b) \in R^{-1}$ [By definition of inverse relation
$\Rightarrow R \subset R^{-1}$
Let $(x, y) \in R^{-1}$
$\Rightarrow(y, x) \in R$ [By definition of inverse relation]
$\Rightarrow(x, y) \in R$ [Since $R$ is symmetric]
$\Rightarrow R^{-1} \subset R$
Thus, $R=R^{-1}$
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