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On $\Bbb Z$ consider the relation $xRy \Leftrightarrow x-y \not\equiv 0 \mod 3$.

Prove (with explanation), whether the relation reflexive, symmetric, antisymmetric transitive is and prove if they are equivalence relation or order relation

I have computed:

1) Reflexive NO

2) Symmetric YES

3) Antisymmetric NO (I'm not sure here)

4) Transitive YES ( I'm not sure here as well)

Is this a good solution? If not, can you explain where the mistake is?

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Your relation can be rewritten as $xRy \Leftrightarrow x \not\equiv y \mod 3$. Note that $4R2$, $2R1$, yet $4 \not R 1$, so $R$ is NOT transitive.

Concerning antisymmetry, assume that $xRy, x \ne y$. It is obvious, then, that $yRx$, which shows that, indeed, $R$ is NOT antisymmetric (because $\equiv$ is), as you suspected. The other two are fine, as well.

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