I am having trouble with this exercise.
Let $O: A \leftrightarrow A$ be a partial order relation over $A$ (reflexive, transitive). Prove that $E: A \leftrightarrow A$, defined as $E = O\cap O^{T}$ (where $O^{T}$ is the transpose or inverse of $O$) is an equivalence relation.
The fact is that I'm trying to use the definitions of a reflexive relation and a transitive relation to define $O$ as a set. I've been trying to define $O$ as: $O = \{(a,b)| aOb \wedge(aOb\wedge bOc\rightarrow aOc)\}$ so as to intersect both $O$ and $O^{T}$ (which, by the way, I am not able to define) and get an equivalence relation. I cannot seem to go much further than defining $O$ (I'd bet it is wrongly defined).
Thanks.