Modulo Multiplication is Associative/Proof 1
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Theorem
Multiplication modulo $m$ is associative:
- $\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m = \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}$
That is:
- $\forall x, y, z \in \Z_m: \paren {x \cdot_m y} \cdot_m z = x \cdot_m \paren {y \cdot_m z}$
Proof
\(\ds \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m\) | \(=\) | \(\ds \eqclass {x y} m \times_m \eqclass z m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {\paren {x y} z} m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {x \paren {y z} } m\) | Integer Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass x m \times_m \eqclass {y z} m\) | Definition of Modulo Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}\) | Definition of Modulo Multiplication |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19.1$: Properties of $\Z_m$ as an algebraic system