Like I said in the last post, this post will prove that a binary relation on a set is an equivalence relation if and only if it is both reflexive and circular. So, assume is an equivalence relation. We have to prove that is both reflexive and circular. Since reflexivity is one of the conditions of being an equivalence relation, we only really have to prove that is circular. Let be elements of , and assume that . We have to show that . By transitivity, , and then by symmetry, , so we are done. Now, for the other direction, assume is both reflexive and circular. We have to prove is an equivalence relation. We already have reflexivity, so we have to prove that is both symmetric and transitive. I will prove this by first proving symmetry, and then using symmetry to prove transitivity. So, for the proof of symmetry, assume . We have to show that . Since is reflexive, we know that . By circularity, we have , so we are done. Now, for transitivity, assume . We have to show that . By circularity, , and then by the symmetry that was just proved, , so we are done. The proof is complete.