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

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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 a congruence relation for multiplication on $\C$.


Proof

Note that by Equivalence Relation Examples: Equal Fourth Powers over Complex Numbers, $\RR$ is an equivalence relation.

It remains to be shown that it is a congruence.


Let $w_1, w_2, z_1, z_2 \in \C$ such that:

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

Then:

\(\ds w_1^4\) \(=\) \(\ds w_2^4\)
\(\, \ds \land \, \) \(\ds z_1^4\) \(=\) \(\ds z_2^4\)
\(\ds \leadsto \ \ \) \(\ds w_1^4 \times z_1^4\) \(=\) \(\ds w_2^4 \times z_2^4\)
\(\ds \leadsto \ \ \) \(\ds \paren {w_1 \times z_1}^4\) \(=\) \(\ds \paren {w_2 \times z_2}^4\)
\(\ds \leadsto \ \ \) \(\ds w_1 \times z_1\) \(\RR\) \(\ds w_2 \times z_2\)

$\blacksquare$


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