Congruence Relation/Examples/Equal Sine of pi x over 6 on Integers for Multiplication

Example of Congruence 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 not a congruence relation for multiplication on $\Z$.

Proof

 * Proof by Counterexample

Note that by Equivalence Relation Examples: Equal $\sin \dfrac {\pi x} 6$ on Integers, $\RR$ is an equivalence relation.

However:

But:
 * $\sin \dfrac {\paren {1 \times 4} \pi} 6 = \dfrac {\sqrt 3} 2$

while:
 * $\sin \dfrac {\paren {5 \times 2} \pi} 6 = -\dfrac {\sqrt 3} 2$

So while we have:
 * $\paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2}$

where $x_1 = 1$, $x_2 = 5$, $y_1 = 4$, $y_2 = 2$

we have:
 * $\paren {x_1 \times y_1} \not \mathrel \RR \paren {x_2 \times y_2}$