Congruence Relation/Examples/Equal Fourth Powers over Complex Numbers for Addition

Example of Congruence Relation
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 $\C$.

Proof

 * Proof by Counterexample

${w_1 \mathrel \RR w_2} \land \paren {z_1 \mathrel \RR z_2}$

Then:

But:
 * $\paren {i + -i}^4 = 0$

while:
 * $\paren {i + 1}^4 = -4$

So while we have:
 * $\paren {w_1 \mathrel \RR w_2} \land \paren {z_1 \mathrel \RR z_2}$

where $w_1 = i$, $w_2 = 1$, $z_1 = -i$, $z_2 = -i$

we have:
 * $\paren {w_1 + z_1} \not \mathrel \RR \paren {w_2 + z_2}$