Equivalence Relation/Examples/Equal Fourth Powers over Complex Numbers/Proof 2
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Example of Equivalence Relation
Let $\C$ denote the set of complex numbers.
Let $\RR$ denote the relation on $\C$ defined as:
- $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$
Then $\RR$ is an equivalence relation.
Proof
We have that $\RR \subseteq \R \times \R$ is the relation induced by $z^4$:
- $\tuple {z, w} \in \RR \iff z^4 = w^4$
The result follows from Relation Induced by Mapping is Equivalence Relation.
$\blacksquare$