In: Advanced Math
Let R be a relation on a set that is reflexive and symmetric but not transitive? Let R(x) = {y : x R y}. [Note that R(x) is the same as x / R except that R is not an equivalence relation in this case.] Does the set A = {R(x) : x ∈ A} always/sometimes/never form a partition of A? Prove that your answer is correct. Do not prove by examples.