A relation $\mathcal{R}$ on $A$ is antisymmetric iff $\forall x, y \in A$ $(x\mathcal{R}y \land y\mathcal{R}x)\rightarrow (x=y)$ For a finite set $A$ with $n$ elements, the number of possible antisymmetric relations is $2^n 3^{\frac{n^2-n}{2}}$ out of the $2^{n^2}$ total possible relations.

Antisymmetric is not the same thing as ``not symmetric'', as it is possible to have both at the same time. However, a relation $\mathcal{R}$ that is both antisymmetric and symmetric has the condition that $ x\mathcal{R}y \Rightarrow x=y $ There are only $2^n$ such possible relations on $A$

An example of an antisymmetric relation on $A = \{\circ, \times, \star\}$ would be $\mathcal{R} = \{(\star,\star),(\times,\circ),(\circ,\star),(\star,\times)\}$ One relation that isn't antisymmetric is $\mathcal{R} = \{ (\times,\circ), (\star, \circ), (\circ,\star) \} $ because we have both $\star \mathcal{R} \circ$ and $\circ \mathcal{R} \star$ but $\circ \not = \star$