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.
Consider matrix which has ones on diagonal and zeros on other places. Symmetric property: $\forall a,b\in X$ $aRb\implies bRa$. Antisymmetric property: $\forall a,b\in X$ ($aRb \land bRa)\implies a=b$. So consider relation $R=\{(x_1,x_1),(x_2,x_2)...(x_n,x_n)\}$ s.t. $x_i\in X$ As you see both properties are hold, so we get matrix - $a_{ij}=1$ for $i=j$ and $a_{ij}=0$ for $i\neq j$.