Equivalence Relation/Examples/Equal Fourth Powers over Complex Numbers/Proof 2

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.

$\blacksquare$