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

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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 1

Checking in turn each of the criteria for equivalence:


Reflexivity

Let $x \in \Z$.

Then:

$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi x} 6$

Thus:

$\forall x \in \Z: x \mathrel \RR x$

and $\RR$ is seen to be reflexive.

$\Box$


Symmetry

\(\ds x\) \(\RR\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \sin \dfrac {\pi x} 6\) \(=\) \(\ds \sin \dfrac {\pi y} 6\)
\(\ds \leadsto \ \ \) \(\ds \sin \dfrac {\pi y} 6\) \(=\) \(\ds \sin \dfrac {\pi x} 6\)
\(\ds \leadsto \ \ \) \(\ds y\) \(\RR\) \(\ds x\)


Thus $\RR$ is seen to be symmetric.

$\Box$


Transitivity

\(\ds x\) \(\RR\) \(\ds y\)
\(\, \ds \land \, \) \(\ds y\) \(\RR\) \(\ds z\)
\(\ds \leadsto \ \ \) \(\ds \sin \dfrac {\pi x} 6\) \(=\) \(\ds \sin \dfrac {\pi y} 6\)
\(\, \ds \land \, \) \(\ds \sin \dfrac {\pi y} 6\) \(=\) \(\ds \sin \dfrac {\pi z} 6\)
\(\ds \leadsto \ \ \) \(\ds \sin \dfrac {\pi x} 6\) \(=\) \(\ds \sin \dfrac {\pi z} 6\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\RR\) \(\ds z\)

Thus $\RR$ is seen to be transitive.

$\Box$


$\RR$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

$\blacksquare$


Proof 2

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.

$\blacksquare$


Sources

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