# 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.

## Proof

That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.

From Number of Equivalence Classes we have:

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.

$\blacksquare$