Say I have a set = {1,2,3}.

I am trying to think about how I could define a set on X which is antisymmetric but not symmetric. 

At first I had thought the set would be Z = {(1,1),(2,2),(3,3)} but am I correct in thinking that this is symmetric? 

Would the set Z = {(1,1),(2,2),(3,3),(1,2),(2,3)} be both antisymmetric, but not symmetric? Since the (1,1),(2,2) and (3,3) make it antisymmetric, but the fact that (2,1) and (3,2) are missing making it not symmetric?

Thanks