## You are here

Homeantisymmetric

## Primary tabs

# 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 $\circ\not=\star$

## Mathematics Subject Classification

03E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections