Absolute Value Function on Integers induces Equivalence Relation
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Theorem
Let $\Z$ be the set of integers.
Let $\mathcal R$ be the relation on $\Z$ defined as:
- $\forall x, y \in \Z: \left({x, y}\right) \in \mathcal R \iff \left\vert{x}\right\vert = \left\vert{y}\right\vert$
where $\left\vert{x}\right\vert$ denotes the absolute value of $x$.
Then $\mathcal R$ is an equivalence relation.
Proof
$\mathcal R$ is shown to be an equivalence relation thus:
Reflexivity
- $\forall x \in \Z: \left\vert{x}\right\vert = \left\vert{x}\right\vert$
Thus $\mathcal R$ is reflexive.
$\Box$
Symmetry
- $\forall x, y \in \Z: \left\vert{x}\right\vert = \left\vert{y}\right\vert \implies \left\vert{y}\right\vert = \left\vert{x}\right\vert$
Thus $\mathcal R$ is symmetric.
$\Box$
Transitive
- $\forall x, y, z \in \Z: \left\vert{x}\right\vert = \left\vert{y}\right\vert \land \left\vert{y}\right\vert = \left\vert{z}\right\vert \implies \left\vert{x}\right\vert = \left\vert{z}\right\vert$
Thus $\mathcal R$ is transitive.
$\Box$
Thus, by definition, $\mathcal R$ is an equivalence relation.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 11$: Example $11.1$