Equivalence Relation/Examples/Equal Fourth Powers over Complex Numbers
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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 1
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.
$\Box$
Symmetry
\(\ds z\) | \(\RR\) | \(\ds w\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^4\) | \(=\) | \(\ds w^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds w^4\) | \(=\) | \(\ds z^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds w\) | \(\RR\) | \(\ds z\) |
Thus $\RR$ is seen to be symmetric.
$\Box$
Transitivity
\(\ds z_1\) | \(\RR\) | \(\ds z_2\) | ||||||||||||
\(\, \ds \land \, \) | \(\ds z_2\) | \(\RR\) | \(\ds z_3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds {z_1}^4\) | \(=\) | \(\ds {z_2}^4\) | |||||||||||
\(\, \ds \land \, \) | \(\ds {z_2}^4\) | \(=\) | \(\ds {z_3}^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds {z_1}^4\) | \(=\) | \(\ds {z_3}^4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z_1\) | \(\RR\) | \(\ds z_3\) |
Thus $\RR$ is seen to be transitive.
$\Box$
$\RR$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
$\blacksquare$
Proof 2
We have that $\RR \subseteq \R \times \R$ is the relation induced by $z^4$:
- $\tuple {z, w} \in \RR \iff z^4 = w^4$
The result follows from Relation Induced by Mapping is Equivalence Relation.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.1 \ \text{(a)}$