Equivalence Relation/Examples/Non-Equivalence/People of Different Age

Example of Relation which is not Equivalence

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 { the age of $x$ and $y$ on their last birthdays was not the same}$

Then $\sim$ is not an equivalence relation.

Proof

$\sim$ is antireflexive, as everybody is the same age as themselves.

$\sim$ is symmetric, as two people are either the same age or they are not.

$\sim$ is not transitive, because if $a \sim b$ and $b \sim c$, it is impossible to say whether $a \sim c$.

$\blacksquare$

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Example $\text{A}.2$