Let $R$ be a transitive relation on a set $A$. Define another relation, $S$, such that, for any $x,y \in A$, $Sxy$ iff $Ryx$. Moreover, let $S$ be irreflexive.
Prove: $S$ is asymmetric on $A$. (Hint: Try assuming that $S$ is not asymmetric and derive a contradiction.)
Here's what I have so far:
Assume $x \in A$.
Then Not $Sxx$ ($S$ is irreflexive)
Assume $x,y,z \in A$
Then $Rxy$ (Definition of $S$)
So $Rxy$ and $Ryz$ (SC conjuntion)
Therefore, $Rxz$ ($R$ is transitive)
Assume $x,y \in A$
I don't understand how to prove $S$ is asymmetric by contradiction.