Modulo Multiplication is Well-Defined/Proof 2
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Theorem
The multiplication modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule:
- $\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$
is a well-defined operation.
That is:
- If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a x \equiv b y \pmod m$.
Proof
The equivalence class $\eqclass a m$ is defined as:
- $\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$
that is, the set of all integers which differ from $a$ by an integer multiple of $m$.
Thus the notation for multiplication of two residue classes modulo $z$ is not usually $\eqclass a m \times_m \eqclass b m$.
What is more normally seen is:
- $a b \pmod m$
Using this notation:
\(\ds a\) | \(\equiv\) | \(\ds b\) | \(\ds \pmod m\) | |||||||||||
\(\, \ds \land \, \) | \(\ds c\) | \(\equiv\) | \(\ds d\) | \(\ds \pmod m\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a \bmod m\) | \(=\) | \(\ds b \bmod m\) | Definition of Congruence Modulo Integer | ||||||||||
\(\, \ds \land \, \) | \(\ds c \bmod m\) | \(=\) | \(\ds d \bmod m\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds b + k_1 m\) | for some $k_1 \in \Z$ | ||||||||||
\(\, \ds \land \, \) | \(\ds c\) | \(=\) | \(\ds d + k_2 m\) | for some $k_2 \in \Z$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a c\) | \(=\) | \(\ds \paren {b + k_1 m} \paren {d + k_2 m}\) | Definition of Multiplication | ||||||||||
\(\ds \) | \(=\) | \(\ds b d + b k_2 m + d k_1 m + k_1 k_2 m^2\) | Integer Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds b d + \paren {b k_2 + d k_1 + k_1 k_2 m} m\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a c\) | \(\equiv\) | \(\ds b d\) | \(\ds \pmod m\) | Definition of Modulo Multiplication |
$\blacksquare$
Warning
This result does not hold when $a, b, x, y, m \notin \Z$.
Let $z \in \R$ be a real number.
Let:
- $a \equiv b \pmod z$
and:
- $x \equiv y \pmod z$
where $a, b, x, y \in \R$.
Then it does not necessarily hold that:
- $a x \equiv b y \pmod z$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.2$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $17$