# Equivalence Relation/Examples/Non-Equivalence/People with Common Ancestor

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## Example of Equivalence 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 { $x$ and $y$ have an ancestor in common}$

Then $\sim$ is not an equivalence relation.

## Proof

We have that $\sim$ is reflexive and symmetric.

But $\sim$ is not transitive:

Let $a$ and $b$ be half-siblings, whose mother is $d$ and whose fathers are $e$ and $f$ respectively.

Then $a \sim b$ as they share an ancestor $d$.

Let $a$ and $g$ be full siblings, whose mother is $i$ and whose father is $e$.

Then $g \sim a$ as they share father $e$ and mother $i$.

But $b$ and $g$ do not have the same father or mother, and it is possible that they share no ancestor.

And in that case:

- $g \nsim b$

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $1 \ \text {(iii)}$