Congruence Relation/Examples
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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 $\Z$.