# Absolute Value induces Equivalence not Compatible with Integer Addition

## 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

However, consider that:

 $\displaystyle \size {-1} = \size 1$ $\leadsto$ $\displaystyle -1 \mathop \RR 1$ $\displaystyle \size 2 = \size 2$ $\leadsto$ $\displaystyle 2 \mathop \RR 2$

$-1 + 2 = 1$

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.

$\blacksquare$