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 $\RR$ denote the relation on $\Z$ defined as:
- $\forall x, y \in \Z: x \mathrel \RR y \iff x + y \text { is divisible by } 3$
Then $\RR$ is not an equivalence relation.
Proof
Let $x = 1$.
Then:
- $x + x = 2$
and so $x + x$ is not divisible by $3$.
Thus $\RR$ is seen to be non-reflexive.
Hence by definition $\RR$ 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 $4$