Equivalence Relation/Examples/Non-Equivalence/People of Different Age
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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 $\text{A}.2$