Let X be a non empty set and P(X) be the set of all subsets of X. For $\mathrm{A},\mathrm{B}\in \mathrm{P}\left(\mathrm{X}\right)$, power set of X, ARB if $\mathrm{A}\cap \mathrm{B}\ne \mathrm{\varphi }$ then the relation is 

$$\begin{gathered} \mathrm{A}\cap \mathrm{A}=\mathrm{A}=\varphi \text{ for a non empty set }\mathrm{A}\text{. So the relation }\mathrm{R}\text{ is not reflexive.} \\ \text{If }\mathrm{A}\cap \mathrm{A}=\varphi \text{ then }\mathrm{B}\cap \mathrm{A}=\varphi \text{ so the solution }\mathrm{R}\text{ is symmetric. } \\ \text{ If }X=1,2,3,4\text{ and }\mathrm{A}=1,\mathrm{B}=2,3\text{ and }\mathrm{C}=1,4\text{ then }\mathrm{A}\cap \mathrm{A}=\varphi ,\mathrm{B}\cap \mathrm{C}=\varphi \text{ and }\mathrm{A}\cap \mathrm{C}=1 \\ \text{Thus }\mathrm{ARB},\mathrm{BRC}\text{ but }\mathrm{A}\text{ is not related }\mathrm{C} \\ \text{ Hence }\mathrm{R}\text{ is not transitive} \end{gathered}$$

R is only symmetric