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
Best Answer
Suppose $R$ is a relation on a set $E$ which is both symmetric and antisymmetric.
Take $a\in E$. Assume you can find $b\in E$ such that $aRb$. By symmetry you get $bRa$. Hence by antisymmetry $a=b$. The same thing holds with $bRa$.
Whence an element is, at most, in relation with itself. So the diagonal set and its subsets are the only example of relation being both symmetric and antisymmetric.