Absolute Value induces Equivalence Compatible with Integer Multiplication

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 Absolute Value Function is Completely Multiplicative:

\(\ds \size {x_1 y_1}\)

\(=\)

\(\ds \size {x_1} \size {y_1}\)

\(\ds \)

\(=\)

\(\ds \size {x_2} \size {y_2}\)

\(\ds \)

\(=\)

\(\ds \size {x_2 y_2}\)

That is:

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

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

$\blacksquare$

Also see

• Absolute Value induces Equivalence not Compatible with Integer Addition

Sources

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