# Absolute Value induces Equivalence Compatible with Integer Multiplication

## 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 a congruence relation for integer multiplication.

## Proof

From Absolute Value Function on Integers induces Equivalence Relation, $\mathcal R$ is an equivalence relation.

Let:

$\left\vert{x_1}\right\vert = \left\vert{x_2}\right\vert$
$\left\vert{y_1}\right\vert = \left\vert{y_2}\right\vert$

Then by definition of absolute value:

 $\displaystyle \left\vert{x_1 y_1}\right\vert$ $=$ $\displaystyle \left\vert{x_1}\right\vert \left\vert{y_1}\right\vert$ $\displaystyle$ $=$ $\displaystyle \left\vert{x_2}\right\vert \left\vert{y_2}\right\vert$ $\displaystyle$ $=$ $\displaystyle \left\vert{x_2 y_2}\right\vert$

That is:

$\left({x_1 y_1, x_2 y_2}\right) \in \mathcal R$

That is, $\mathcal R$ is a congruence relation for integer multiplication.

$\blacksquare$