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.

You can't “give an example” for antisimmetry. You could give an example for failure of antisymmetry.
How can you show this is antisymmetric? Let's stretch it a bit and suppose we have also proved it is transitive. Then it should be an order relation and the pairs tell us that


*

*$a$ precedes $b$

*$a$ precedes $c$

*$a$ precedes $d$

*$d$ precedes $b$

*$d$ precedes $c$


and no other relation between distinct elements holds. Now we can see that all pairs necessary for the order relation above are indeed present. So the relation $R$ is indeed antisymmetric and transitive.
This is the Hasse diagram for the relation.

