Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined

Theorem

Let $\beta$ denote the relation defined on the integers $\Z$ by:

$\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$

We have that $\beta$ is an equivalence relation.

Let the $+_\beta$ operator ("addition") on the $\beta$-equivalence classes be defined as:

$\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$

Then such an operation is not well-defined.

We have:

$\ds \eqclass 1 \beta$

$=$

$\ds \eqclass 4 \beta$

$\ds \eqclass 2 \beta$

$=$

$\ds \eqclass 3 \beta$

Thus:

$\ds \eqclass 0 \beta$

$=$

$\ds \eqclass 5 \beta$

$\ds $

$=$

$\ds \eqclass {1 + 4} \beta$

$\ds $

$=$

$\ds \eqclass 1 \beta + \eqclass 4 \beta$

$\ds $

$=$

$\ds \eqclass 1 \beta + \eqclass 1 \beta$

$\ds $

$=$

$\ds \eqclass 2 \beta$

Hence the result.

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $9 \ \text {(i)}$