$S$ is the set of Integers, $a,b$ both belongs to $S$, $a\sim b$ if both $a>b$ and $b>a$.

Question

The original question was

"Is the relation $\sim$ is a equivalence relation?"

I can mathematically figure it out that this is not.

Since $a$ is not related to $a$, as $a$ satisfies neither of the conditions.
Basically no two pair of integers satisfies this.

My question is,

1. Is my approach correct?
2. Since no element is mapped to no other element, can we even call it a relation?

Answer 1

No, it is not an equivalence relation since, as you have correctly determined, it is not reflexive.

It is the binary relation $\emptyset$ since $a\not{\sim}b$ for all $a, b\in \Bbb Z$, because $\emptyset\in \mathcal P(\Bbb Z\times \Bbb Z)$.