Equivalence Relation/Examples/Equal Sine of pi x over 6 on Integers/Proof 2

Example of Equivalence Relation

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then $\RR$ is an equivalence relation.

Proof

We have that $\RR \subseteq \R \times \R$ is the relation induced by $\sin \dfrac {x \pi} 6$:

$\tuple {x, y} \in \RR \iff \sin \dfrac {x \pi} 6 = \sin \dfrac {y \pi} 6$

The result follows from Relation Induced by Mapping is Equivalence Relation.

$\blacksquare$