Let R be a reflexive relation on a nonempty set X. The asymmetric part of R is defined as the relation $P_r$ on X as $xP_ry$ iff $xRy$ but not $yRx$. The relation $I_r$ = $R\setminus P_r$ on X is then called the symmetric part of R.
- Show that if R is transitive , so are $P_r$ and $I_r$. ( Question from maths for economists by Efe A. Ok)
My attempt : I tried using the definition of transitivity i.e. if $xRy$ and $yRz$ then $xRz$ However to use it properly I had to use cases where $zRx$ belonged to R and also a case where it did not belong to R. However I could not prove it even when using this approach.
Please let me know how to proceed.