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

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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:

\(\ds \sin \dfrac \pi 6\) \(=\) \(\ds \sin \dfrac {5 \pi} 6\) \(\ds = \dfrac 1 2\)
\(\, \ds \land \, \) \(\ds \sin \dfrac {2 \pi} 6\) \(=\) \(\ds \sin \dfrac {4 \pi} 6\) \(\ds = \dfrac {\sqrt 3} 2\)
\(\ds \leadsto \ \ \) \(\ds 1\) \(\RR\) \(\ds 5\)
\(\, \ds \land \, \) \(\ds 2\) \(\RR\) \(\ds 4\)


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:

$\paren {x_1 + y_1} \not \mathrel \RR \paren {x_2 + y_2}$

$\blacksquare$


Sources