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

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