Mappings Between Residue Classes

Theorem

Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).

Let $\phi: \Z_m \to \Z_n$ be a mapping given by:

$\map \phi {\eqclass x m} = \eqclass x n$

Then $\phi$ is well defined if and only if $m$ is a divisor of $n$.

Proof

For $\phi$ to be well defined, we require that:

$\forall x, y \in \Z_m: \eqclass x m = \eqclass y m \implies \map \phi {\eqclass x m} = \map \phi {\eqclass y m}$

Now:

$\eqclass x m = \eqclass y m \implies x - y \divides m$

For $\map \phi {\eqclass x m} = \map \phi {\eqclass y m}$ we require that:

$\eqclass x n = \eqclass y n \implies x - y \divides n$

Thus $\phi$ is well defined if and only if:

$x - y \divides m \implies x - y \divides n$

That is, if and only if $m \divides n$.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 19 \alpha$