PlanetMath: antisymmetric

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antisymmetric

(Definition)

A relation $\mathcal{R}$ on is antisymmetric iff $\forall x, y \in A$ , $(x\mathcal{R}y \land y\mathcal{R}x)\rightarrow (x=y)$ . For a finite set with 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 such possible relations on .

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 $\circ \not = \star$

"antisymmetric" is owned by aoh45. [ full author list (2) | owner history (1) ]

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Other names:

antisymmetry

Attachments:: example of antisymmetric (Example) by Algeboy

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Cross-references: symmetric, finite set, iff, relation
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This is version 10 of antisymmetric, born on 2002-02-02, modified 2005-02-28.
Object id is 1666, canonical name is Antisymmetric.
Accessed 9570 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

Pending Errata and Addenda

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