Sets And Relations

Let \(P(S)\) denote the power set of \(S=\{1,2,3, \ldots ., 10\}\). Define the relations \(R_{1}\) and \(R_{2}\) on \(P(S)\) as \(\mathrm{AR}_{1} \mathrm{~B}\) if \(\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \cap \mathrm{A}^{\mathrm{c}}\right)=\emptyset\) and \(\mathrm{AR}_{2} \mathrm{~B}\) if \(\mathrm{A} \cup \mathrm{B}^{\mathrm{c}}=\mathrm{B} \cup \mathrm{A}^{\mathrm{c}}, \forall \mathrm{A}, \mathrm{B} \in \mathrm{P}(\mathrm{S})\). Then :