# Equivalence Relation/Examples/Non-Equivalence/Sum of Integers is Divisible by 3

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## Example of Relation which is not Equivalence

Let $\Z$ denote the set of integers.

Let $\mathcal R$ denote the relation on $\Z$ defined as:

- $\forall x, y \in \Z: x \mathrel {\mathcal R} y \iff x + y \text { is divisible by $3$}$

Then $\mathcal R$ is not an equivalence relation.

## Proof

Let $x = 1$.

Then:

- $x + x = 2$

and so $x + x$ is not divisible by $3$.

Thus $\mathcal R$ is seen to be non-reflexive.

Hence by definition $\mathcal R$ is not an equivalence relation.

$\blacksquare$

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: Problem Set $\text{A}.3$: $20$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Exercise $4$