Equivalence Relation/Examples/Non-Equivalence/Divisor Relation

Example of Relation which is not Equivalence

Let $\Z_{>0}$ 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 $1 \ \text {(iv)}$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): relation: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): relation: 1.