Symmetric Relation/Examples

Examples of Symmetric Relations

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:

$b \not \sim a$

Cousin 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 { $x$ is a cousin of $y$}$

Then $\sim$ is a symmetric relation.

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.