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

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 addition 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:
 * $\sin \dfrac {\paren {1 + 2} \pi} 6 = 1$

while:
 * $\sin \dfrac {\paren {4 + 5} \pi} 6 = -1$

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

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

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