antisymmetric

A relation $ℛ$ on $A$ is antisymmetric iff$\forall x,y\in A$, $(xℛy\wedge yℛx)\to (x=y)$.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 possiblerelations.

Antisymmetric is not the same thing as “not symmetric”, as it is possibleto have both at the same time. However, a relation $ℛ$ that is bothantisymmetric and symmetric has the condition that $xℛy\Rightarrow x=y$.There are only $2^n$ such possible relations on $A$.

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