Equivalence Relation/Examples/Non-Equivalence/Is the Mother Of

Example of Relation which is not Equivalence
Let $P$ denote 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 the mother of $y$}$

Then $\sim$ is not an equivalence relation.

The same applies (trivially) to the relation:
 * $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ is the father of $y$}$

Proof
For a start, no person can be his or her own mother, so:
 * $\forall x: x \nsim x$

So $\sim$ is not reflexive.

Then:
 * If $x \sim y$ then $y$ is the son or daughter of $x$.

So $\sim$ is not symmetric.

Then:
 * if $x \sim y$ and $y \sim z$ it follows that $x$ is the grandmother of $z$, not his or her mother.

So $\sim$ is not transitive.