Absolute Value Function on Integers induces Equivalence Relation

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 $11.1$