1. The problem statement, all variables and given/known data Assume a relation [itex]P[/itex] that is asymmetric 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 complete 2. Relevant equations Asymmetry: [itex]xRy \rightarrow \neg (yRx) [/itex] Now, I think I got a proof, but I am not sure how well I can express it. So my problem is both about contents and style. 3. The attempt at a solution PROOF: Let [itex]x[/itex], [itex]y[/itex] be arbitrary. Assume that [itex]xRy[/itex] is false. Thus, by definition of [itex]R[/itex], it follows that [itex]yPx[/itex]. Since [itex]P[/itex] is an asymmetric relation, if follows that [itex]xPy[/itex] is false. Therefore, since [itex]x[/itex] and [itex]y[/itex] are arbitrary, it follows from the definition of [itex]R[/itex] that the relation is complete. Does it look perspicuous or there is something not clear? Maybe the last sentence? Thanks a lot in advance.