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**oldguynewstudent** Let S = {1,2,3,4,5}. On P(S) define the relation $\displaystyle \rho$ as follows: A$\displaystyle \rho$B iff A$\displaystyle \cap$B = $\displaystyle \emptyset$. Is $\displaystyle \rho$ reflexive? Is $\displaystyle \rho$ symmetric? Is $\displaystyle \rho$ transitive?

Since there are elements in P(S) that do not equal $\displaystyle \emptyset$ when intersecting themself then $\displaystyle \rho$ is not reflexive.

If A$\displaystyle \cap$B is empty then B$\displaystyle \cap$A is empty so $\displaystyle \rho$ is symmetric.

{1,2} $\displaystyle \cap$ {3,4} is empty. And {3,4} $\displaystyle \cap$ {2,5} is empty. But {1,2} $\displaystyle \cap$ {2,5} = {2}. So $\displaystyle \rho$ is not transitive.