Absolute Value Function on Integers induces Equivalence Relation

Theorem

Let $\Z$ be the set of integers.

Let $\RR$ be the relation on $\Z$ defined as:

$\forall x, y \in \Z: \tuple {x, y} \in \RR \iff \size x = \size y$

where $\size x$ denotes the absolute value of $x$.

Then $\RR$ is an equivalence relation.

Proof

$\RR$ is shown to be an equivalence relation thus:

Reflexivity

$\forall x \in \Z: \size x = \size x$

Thus $\RR$ is reflexive.

$\Box$

Symmetry

$\forall x, y \in \Z: \size x = \size y \implies \size y = \size x$

Thus $\RR$ is symmetric.

$\Box$

Transitive

$\forall x, y, z \in \Z: \size x = \size y \land \size y = \size z \implies \size x = \size z$

Thus $\RR$ is transitive.

$\Box$

Thus, by definition, $\RR$ is an equivalence relation.

$\blacksquare$

Sources

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