Definition:Symmetry (Relation)

From ProofWiki
Jump to navigation Jump to search

Definition

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


Symmetric

$\RR$ is symmetric if and only if:

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


Asymmetric

$\RR$ is asymmetric if and only if:

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


Antisymmetric

$\RR$ is antisymmetric if and only if:

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

that is:

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


Non-symmetric

$\RR$ is non-symmetric if and only if it is neither symmetric nor asymmetric.


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.


Linguistic Note

The word symmetry comes from Greek συμμετρεῖν (symmetría) meaning measure together.


Also see

  • Results about symmetric relations can be found here.