Modulo Addition is Commutative

Theorem

$\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$

From the definition of modulo addition, this is also written:

$\forall m \in \Z: \forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m = \eqclass y m +_m \eqclass x m$

Hence:

$\ds \eqclass x m +_m \eqclass y m$

$=$

$\ds \eqclass {x + y} m$

$\ds $

$=$

$\ds \eqclass {y + x} m$

$\ds $

$=$

$\ds \eqclass y m +_m \eqclass x m$

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.7$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19.1$: Properties of $\Z_m$ as an algebraic system