Modulo Addition is Associative

Theorem

Addition modulo $m$ is associative:

$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \R_m: \left({\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m}\right) +_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m +_m \left({\left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m}\right)$

where $\R_m$ is the set of all residue classes modulo $m$.

That is:

$\forall x, y, z \in \R: \left({x + y}\right) + z \equiv x + \left({y + z}\right) \pmod m$

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m}\right) +_m \left[\!\left[{z}\right]\!\right]_m\)	\(=\)	\(\displaystyle \left[\!\left[{x + y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of addition modulo $m$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left({x + y}\right) + z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of addition modulo $m$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x + \left({y + z}\right)}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Addition of Numbers is Associative
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y + z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of addition modulo $m$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m +_m \left({\left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of addition modulo $m$

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $2.6$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19.1$