Find total number of relations that are equivalence as well as partial order set 
Find total number of relations that are equivalence as well as partial order set. Assume set contains total $n$ elements.


My attempt:
As equivalence relation has property reflexive, symmetric and transitive and POSET has property reflexive, antisymmetric and transitive.
Symmetric and antisymmetric will cancle each other except reflexive relation and transitive relation can case antisymmetric, so it's also absent on resultant relation.
Therefore, all diagonal element should be present to satisfy conditions of equivalence as well as POSET.
So, it's independent of number of total number of element of set and answer will be $1$.

Can you explain in formal way, please?


I will denote $\sim$ such a relation.
If $a\sim b$, then $b\sim a$ (symmetry), and then $a=b$ (antisymmetry).
The order is partial, but each time you can compare two elements with each other, they must be the same.
So the only element you can compare with $a$ (for any $a$) is $a$ itself.
Therefor there is only one such relation, which is trivial:
$$\forall a,b\quad a\sim b \iff a=b.$$
