Question

If R is a symmetric relation on a set A, then write a relation between R and R⁻¹.

Answer 1

Here, R is symmetric on the set A.

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow \left(b,a\right)\in R\left[\mathrm{Since}R\mathrm{is}\mathrm{symmetric}\right] \\ \Rightarrow \left(a,b\right)\in R^{-1}\left[\mathrm{By}\mathrm{definition}\mathrm{of}\mathrm{inverse}\mathrm{relation}\right] \\ \Rightarrow R\subset R^{-1} \\ \mathrm{Let}\left(x,y\right)\in R^{-1} \\ \Rightarrow \left(y,x\right)\in R\left[\mathrm{By}\mathrm{definition}\mathrm{of}\mathrm{inverse}\mathrm{relation}\right] \\ \Rightarrow \left(x,y\right)\in R\left[\mathrm{Since}R\mathrm{is}\mathrm{symmetric}\right] \\ \Rightarrow R^{-1}\subset R \\ \mathrm{Thus},R=R^{-1} \end{gathered}$$