Modulo Addition is Well-Defined/Proof 2

Theorem

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

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

Let $\eqclass a m$ denote the equivalence class on $\Z_m$, for some $a \in \Z$.

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 $a + x \equiv b + y \pmod m$.

Proof

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

$\blacksquare$

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences: $\text{(i)}$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.2$
1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.10$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 18.4$: Congruence classes
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $17$