Symmetric Relation/Examples/Is a Brother of

Example of Symmetric 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:
 * $b \not \sim a$