Equivalence Relation/Examples/Equal Fourth Powers over Complex Numbers

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
Checking in turn each of the criteria for equivalence:

Reflexivity
Let $z \in \C$.

Then:
 * $z^4 = z^4$

Thus:
 * $\forall z \in \C: z \mathrel \RR z$

and $\RR$ is seen to be reflexive.

Symmetry
Thus $\RR$ is seen to be symmetric.

Transitivity
Thus $\RR$ is seen to be transitive.

$\RR$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.