Definition:Antisymmetric Relation

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.


Definition 1

$\RR$ is antisymmetric if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$

that is:

$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$


Definition 2

$\RR$ is antisymmetric if and only if:

$\tuple {x, y} \in \RR \land x \ne y \implies \tuple {y, x} \notin \RR$


Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation in $V$.


$\RR$ is antisymmetric if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$

that is:

$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$


Also known as

Some sources render antisymmetric relation as anti-symmetric relation.


Antisymmetric and Asymmetric Relations

Note the difference between:

An asymmetric relation, in which the fact that $\tuple {x, y} \in \RR$ means that $\tuple {y, x}$ is definitely not in $\RR$

and:

An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$ but if there are, then it means that $x$ and $y$ have to be the same object.


Examples

Ordering of Integers

The usual ordering $\le$ on the set of integers $\Z$ is antisymmetric:

$\forall x, y \in \Z: \paren {x \le y} \land \paren {y \le x} \iff x = y$


Set Inclusion

The subset relation is antisymmetric:

$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$

where $x$ and $y$ are sets.


Partial Ordering

Let $\preccurlyeq$ be a partial ordering on a set $S$.

Then $\preccurlyeq$ is an antisymmetric relation on $S$.


Also see

  • Results about antisymmetric relations can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Antisymmetric_Relation&oldid=746070"