Symmetric Relation/Examples/Is of Opposite Gender

Example of Symmetric 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.