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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.6$
- 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): $\S 19.1$: Properties of $\Z_m$ as an algebraic system