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

We have from Equivalence Relation Examples: Equal Fourth Powers over Complex Numbers, $\RR$ is an equivalence relation.

However:

\(\ds i^4\)

\(=\)

\(\ds i^4\)

\(\, \ds \land \, \)

\(\ds 1^4\)

\(=\)

\(\ds \paren {-i}^4\)

\(\ds \leadsto \ \ \)

\(\ds i\)

\(\RR\)

\(\ds i\)

\(\, \ds \land \, \)

\(\ds 1\)

\(\RR\)

\(\ds -i\)

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:

$\paren {w_1 + z_1} \not \mathrel \RR \paren {w_2 + z_2}$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.1 \ \text{(a)}$