Definition:Symmetric Relation
Definition
Let $\RR \subseteq S \times S$ 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
$\RR$ is symmetric if and only if it is a subset of its inverse:
- $\RR \subseteq \RR^{-1}$
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 symmetric if and only if:
- $\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$
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:
- $b \not \sim a$
Opposite Gender Relation
Let $P$ be the set of people.
Let $\sim$ be the relation on $P$ defined as:
- $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ of the opposite gender to $y$}$
(This assumes that gender is binary and well-defined.)
Then $\sim$ is a symmetric relation.
However, $\sim$ is antireflexive and antitransitive.
Distance Less than 1
Let $\sim$ be the relation on the set of real numbers $\R$ defined as:
- $x \sim y \iff \size {x - y} < 1$
Then $\sim$ is symmetric and reflexive, but not transitive.
Also see
- Results about symmetric relations can be found here.