1. The problem statement, all variables and given/known data Assume a relation [itex]P[/itex] that is negatively transitive on a set [itex]X[/itex] that is not empty. Define the binary relation [itex]R[/itex] on [itex]X[/itex] by [itex]xRy[/itex] iff [itex] y P x [/itex] is false. Prove that [itex]R[/itex] is transitive. 2. Relevant equations Negative Transitivity: [itex]xPz \rightarrow xPy \vee yPz [/itex] Like in the previous thread (Complete Relation), I think I have a proof, but I am really looking forward to any feedback on the style of this proof. I tried to incorporate the stylistic feedbacks of Michael Redei and pasmith who kindly gave me feedbacks on the previous attempt. 3. The attempt at a solution Proof: Let [itex]x[/itex], [itex]y[/itex] and [itex]z[/itex] be arbitrary and assume [itex]xRy[/itex] and [itex]yRz[/itex]. By definition of [itex]R[/itex] it follows that [itex]yPx[/itex] and [itex]zPy[/itex] are both false. Since [itex]P[/itex] is negatively transitive, this implies that [itex]zPx[/itex] is false. Thus, again by definition of [itex]R[/itex], the falsity of [itex]zPx[/itex] implies [itex]xRz[/itex], which proves that the relation [itex]R[/itex] is transitive.