# antisymmetric

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 $\circ\not=\star$

Title antisymmetric Antisymmetric 2013-03-22 12:15:50 2013-03-22 12:15:50 aoh45 (5079) aoh45 (5079) 14 aoh45 (5079) Definition msc 03E20 antisymmetry Reflexive    Symmetric ExteriorAlgebra SkewSymmetricMatrix