Equivalence relation proof

Question

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.

Answer 1

Hint "Just do it", as they say. By definition, $x E y \equiv x O y \land y O x$, where $O$ is transitive and reflexive. And you have to establish separately each of

1. $x E x$
2. $x E y \to y E x$
3. $x E y \land y E z \to x E z$.

But each of those is immediate given the definition and the condition on $O$ ...