$$ \begin{split}∀x∃y:R(x,y)&∧∀x∀y:(R(x,y)⟹¬R(y,x)) \\ &∧∀x∀y∀z:(R(x,y)∧R(y,z)⟹R(x,z))\\&∧∀x:¬R(x,x) \end{split} $$
Does it have finite models?
Is it satisfiable? If so, give a countable model for it.
Genuinely, I'm unable to understand this question. Somewhere, it explained as : broken in $4$ parts, i.e. $A, B, C, D$. It explained using a graph and conclusion that $B$ is antisymmetric, $C$ is transitive and $D$ is irreflexive relations.
Can you explain little bit please, how do I solve this problem?
If this formula regarding valid/satisfiable, is any counter example. what is meaning of countable model here?
This is a standard sort of homework problem. You are supposed to look at the formula and recognize that all the parts of the formula are axioms for order relations (such as transitivity), and in particular the formula is implied by the axioms for a strict linear order with no maximum element.
That is the general method for many problems of this sort - they simply expect you to draw on your mathematical background to recognize the meaning of the formula.