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.


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

But we have:

$2 \divides 4$


$4 \nmid 2$

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