Modulo Addition is Commutative

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Theorem

Modulo addition is commutative:

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


Proof

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:

\(\displaystyle \eqclass x m +_m \eqclass y m\) \(=\) \(\displaystyle \eqclass {x + y} m\) Definition of Modulo Addition
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {y + x} m\) Commutative Law of Addition
\(\displaystyle \) \(=\) \(\displaystyle \eqclass y m +_m \eqclass x m\) Definition of Modulo Addition

$\blacksquare$


Sources