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.

$\blacksquare$


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