What would be good example of antisymmetry for this relation? We have set
$X=\{a,b,c,d\}$ 
and relation
$R=\{(a,a),(a,b),(a,c),(a,d),(b,b),(c,c),(d,b),(d,c),(d,d)\}$.
It is obvious that it is not symmetric and I suppose that it is antisymmetric but I can't come up with some good example to show it.
I know the rule of antisymmetry $xRy \wedge yRx \rightarrow x = y$, but I am not sure how to apply it on this relation.

A single example will not suffice.   You are required to affirm that a counterexample does not exist.
The relation $\rm R$ will not be antisymmetric if you can find any $x,y$ such that $x\neq y$, $x\operatorname{R} y$, and $y\operatorname{R} x$.   Only if you can be definite that there is no such pair can you assert antisymmetry.
This requires an exhaustive check of the elements of $\rm R$.
