Definition:Symmetric Relation
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Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
Definition 1
$\RR$ is symmetric if and only if:
- $\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$
Definition 2
$\RR$ is symmetric if and only if it equals its inverse:
- $\RR^{-1} = \RR$
Definition 3
$\mathcal R$ is symmetric if and only if it is a subset of its inverse:
- $\mathcal R \subseteq \mathcal R^{-1}$
Examples
Brotherhood Relation
Let $P$ be the set of male people.
Let $\sim$ be the relation on $P$ defined as:
- $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ is a brother of $y$}$
Then $\sim$ is a symmetric relation.
This does not hold if $P$ is the set of all people.
Because if $a$ is male and $b$ are brother and sister, then:
- $a \sim b$
but:
- $b \not \sim a$
Also see
- Results about symmetry of relations can be found here.