Equivalence Relation/Examples/Equal Sine of pi x over 6 on Integers/Proof 1

Proof
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.

Symmetry
Thus $\RR$ is seen to be symmetric.

Transitivity
Thus $\RR$ is seen to be transitive.

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

Hence by definition it is an equivalence relation.