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:
\(\ds w_1^4\) | \(=\) | \(\ds w_2^4\) | ||||||||||||
\(\, \ds \land \, \) | \(\ds z_1^4\) | \(=\) | \(\ds z_2^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds w_1^4 \times z_1^4\) | \(=\) | \(\ds w_2^4 \times z_2^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {w_1 \times z_1}^4\) | \(=\) | \(\ds \paren {w_2 \times z_2}^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds w_1 \times z_1\) | \(\RR\) | \(\ds w_2 \times z_2\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.1 \ \text{(a)}$