# Absolute Value induces Equivalence not Compatible with Integer Addition

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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 not a congruence relation for integer addition.

## Proof

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

However, consider that:

\(\displaystyle \size {-1} = \size 1\) | \(\leadsto\) | \(\displaystyle -1 \mathop \RR 1\) | |||||||||||

\(\displaystyle \size 2 = \size 2\) | \(\leadsto\) | \(\displaystyle 2 \mathop \RR 2\) |

By conventional integer addition:

- $-1 + 2 = 1$

while:

- $1 + 2 = 3$

But it does not hold that:

- $\size 1 = \size 3$

Therefore $\RR$ is not a congruence relation for integer addition.

Hence the result, by Proof by Counterexample.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 11$: Example $11.1$