How can I prove $R^T\ ;R^T$ is transitive if $R$ is transitive.

Question

If $R$ is transitive relation. How can I prove that composition of its transpose is also transitive.

i.e. $R^T\ ;R^T$ is transitive too.

Answer 1

Let $x R^T y$ and $ y R^T z$. Then $ zRy $ and $ yRx$ by definition of the transpose relation. Since $R$ is transitive, that means that $zRx$. Again, by definition of the transpose relation, $xR^Tz$.