 It is important to grasp the differences between a relation \(R\) and its ancestral \(R^*\). Rxy implies \(R^*(x,y)\) (e.g., if Clinton is a father of Chelsea, then Clinton is a forefather of Chelsea), but the converse doesn’t hold (Clinton’s father is a father* of Chelsea, but he is not a father of Chelsea). Indeed, a grasp of the definition of \(R^*\) should leave one able to prove the following easy consequences, many of which correspond to theorems in Begr and  Gg:[11]