Absolute Value induces Equivalence Compatible with Integer Multiplication

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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 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.

$\blacksquare$


Also see


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