Congruence Relation/Examples

Examples of Congruence Relations

Equal Fourth Powers over $\C$ for Multiplication

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

$\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is a congruence relation for multiplication on $\C$.

Equal Fourth Powers over $\C$ for Addition

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

$\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is not a congruence relation for addition on $\C$.

$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Multiplication

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 $\Z$.

$\sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$ over $\Z$ for Addition

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 $\Z$.