Absolute Value Function on Integers induces Equivalence Relation
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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 $11.1$