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 $y$}$

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.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.2$. Equivalence relations: Example $31$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations