Modulo Addition is Well-Defined/Proof 2

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Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$.

The modulo addition operation on $\Z_m$, defined by the rule:

$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

is a well-defined operation.

That is:

If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a + x \equiv b + y \pmod m$.


The equivalence class $\eqclass a m$ is defined as:

$\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$

That is, the set of all integers which differ from $a$ by an integer multiple of $m$.

Thus the notation for addition of two set of integers modulo $m$ is not usually $\eqclass a m +_m \eqclass b m$.

What is more normally seen is $a + b \pmod m$.

Using this notation:

\(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod m\)
\(\, \ds \land \, \) \(\ds c\) \(\equiv\) \(\ds d\) \(\ds \pmod m\)
\(\ds \leadsto \ \ \) \(\ds a \bmod m\) \(=\) \(\ds b \bmod m\) Definition of Congruence Modulo Integer
\(\, \ds \land \, \) \(\ds c \bmod m\) \(=\) \(\ds d \bmod m\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds b + k_1 m\) for some $k_1 \in \Z$
\(\, \ds \land \, \) \(\ds c\) \(=\) \(\ds d + k_2 m\) for some $k_2 \in \Z$
\(\ds \leadsto \ \ \) \(\ds a + c\) \(=\) \(\ds b + d + \paren {k_1 + k_2} m\) Definition of Integer Addition
\(\ds \leadsto \ \ \) \(\ds a + c\) \(\equiv\) \(\ds b + d\) \(\ds \pmod m\) Definition of Modulo Addition