Take a look at this picture:
From what I am reading, antisymmetric means:
$$∀ x ∀ y \,[ R ( x , y ) ∧ R ( y , x ) ⇒ x = y ]$$
However, $(2,1)$ and $(1,2)$, $X\ne Y$. I understand how this is symmetric but how is this antisymmetric? I got this from my professor and my book explains that they are not mutually exclusive.