If R is a symmetric relation on a set A, then write a relation be… - Vidyadip

Question

If $R$ is a symmetric relation on a set $A,$ then write a relation between $R$ and $R^{-1}.$

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Answer

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

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