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 $1 \ \text {(iv)}$