Modulo Addition is Associative

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Theorem

Addition modulo $m$ is associative:

$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m +_m \eqclass y m} +_m \eqclass z m = \eqclass x m +_m \paren {\eqclass y m +_m \eqclass z m}$

where $\Z_m$ is the set of integers modulo $m$.


That is:

$\forall x, y, z \in \Z: \paren {x + y} + z \equiv x + \paren {y + z} \pmod m$


Proof

\(\ds \paren {\eqclass x m +_m \eqclass y m} +_m \eqclass z m\) \(=\) \(\ds \eqclass {x + y} m +_m \eqclass z m\) Definition of Modulo Addition
\(\ds \) \(=\) \(\ds \eqclass {\paren {x + y} + z} m\) Definition of Modulo Addition
\(\ds \) \(=\) \(\ds \eqclass {x + \paren {y + z} } m\) Associative Law of Addition
\(\ds \) \(=\) \(\ds \eqclass x m +_m \eqclass {y + z} m\) Definition of Modulo Addition
\(\ds \) \(=\) \(\ds \eqclass x m +_m \paren {\eqclass y m +_m \eqclass z m}\) Definition of Modulo Addition

$\blacksquare$


Sources

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