Exercise 567Logic, sets and proofs › RelationsHigh schoolAdded 3 Nov 2026

Equivalence relations

#equivalence relation#reflexive#symmetric#transitive

Statement

Show that $x\sim y\iff x\equiv y\pmod5$ on $\mathbb{Z}$ is an equivalence relation.

Hint

Check the three axioms.

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Reflexive. $x-x=0$ is divisible by $5$ .

Symmetric. If $5\mid(x-y)$ , then $5\mid(y-x)$ .

Transitive. If $5\mid(x-y)$ and $5\mid(y-z)$ , then $5\mid(x-y)+(y-z)=x-z$ .

Conclusion. $\equiv\pmod5$ is an equivalence relation.