Absolute Value induces Equivalence Compatible with Integer Multiplication

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Theorem

Let $\Z$ be the set of integers.

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

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

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


Then $\RR$ is a congruence relation for integer multiplication.


Proof

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


Let:

$\size {x_1} = \size {x_2}$
$\size {y_1} = \size {y_2}$

Then by definition of absolute value:

\(\displaystyle \size {x_1 y_1}\) \(=\) \(\displaystyle \size {x_1} \size {y_1}\)
\(\displaystyle \) \(=\) \(\displaystyle \size {x_2} \size {y_2}\)
\(\displaystyle \) \(=\) \(\displaystyle \size {x_2 y_2}\)

That is:

$\paren {x_1 y_1, x_2 y_2} \in \RR$

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

$\blacksquare$


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