Why is this not a equivalence relation in the $\mathbb{Z}$ set?

Question

The relation $xRy$ on the set $\mathbb{Z}$ if $x + y$ is divisible by $3$.

$xRx = x + x = 2x$. In order to $2x$ be divisible by $3$, $x$ itself need to be divisible by $3$, it is not a problem is it?

$xRy = yRx$

$xRy = yRz = xRz$

I think the last two are immediate, so probably my error is in the first.

an equivalence relation must be reflexive for all elements in the set — J. W. Tanner, Dec 28 '20 at 08:05
It is not an equivalence relation. The relation is not reflexive as your observed. It is not transitive either, consider $x=1,y=2,z=1$. — Shubham Johri, Dec 28 '20 at 08:07

Robert Z · Answer 1 · 2020-12-28T09:01:16.263

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Note that $1R2$ ($1+2$ is divisible by $3$) and $2R4$ ($2+4$ is divisible by $3$) do not imply $1R4$ ($1+4$ is not divisible by $3$). Hence transitive property does not hold.

Also reflexive property $xRx$ is not satisfied when $x\equiv 1\pmod{3}$ or $x\equiv 2\pmod{3}$.

edited Dec 28 '20 at 09:01

answered Dec 28 '20 at 08:05

Robert Z

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score 1 · Answer 2 · answered Dec 28 '20 at 09:01

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$xRx$ is not true for $x=1$. Therefore, the relation is not reflexive.

answered Dec 28 '20 at 09:01

5xum

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Why is this not a equivalence relation in the $\mathbb{Z}$ set?

2 Answers2