Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations2 Marks
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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