# Equivalence Relation/Examples/Non-Equivalence/Divisor Relation

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## Example of Relation which is not Equivalence

Let $\Z$ denote the set of (strictly) positive integers.

Let $x \divides y$ denote that $x$ is a divisor of $y$

Then $\divides$ is not an equivalence relation.

## Proof

From Divisor Relation on Positive Integers is Partial Ordering we have that $\divides$ is reflexive and transitive.

But we have:

- $2 \divides 4$

and:

- $4 \nmid 2$

So $\divides$ is not symmetric and therefore not an equivalence relation.

$\blacksquare$

## Sources

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