Parity Multiplication is Associative

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring.


The operation $\times$ is associative:

$\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$


Proof 1

From Isomorphism between Ring of Integers Modulo 2 and Parity Ring:

$\struct {\set {\text{even}, \text{odd} }, +, \times}$ is isomorphic with $\struct {\Z_2, +_2, \times_2}$

the ring of integers modulo $2$.


The result follows from:

Modulo Multiplication is Associative

and:

Isomorphism Preserves Associativity.

$\blacksquare$


Proof 2

Let $a, b, c \in R$.

That is, $a, b, c$ are all either $\text{even}$ or $\text{odd}$.


By definition of odd:

$\text{odd} = 2 m + 1$

for some $m \in \Z$.

By definition of even:

$\text{even} = 2 n + 0$

for some $n \in \Z$.

Thus we can define the mapping $f: R \to \Z$ as:

$\forall x \in R: \map f x := \begin{cases} 0 & : x \text { is even} \\ 1 & : x \text { is odd} \end{cases}$


Thus an element of $R$ can be expressed as an arbitrary integer of the form:

$x = 2 k + \map f x$

where:

$k \in \Z$ is an integer
$\map f x$ is either $0$ or $1$ according to whether $x$ is even or odd.


Then:

\(\ds \paren {a \times b} \times c\) \(=\) \(\ds \paren {\paren {2 r + \map f a} \paren {2 s + \map f b} } \paren {2 t + \map f c}\) where $r, s, t \in \Z$
\(\ds \) \(=\) \(\ds \paren {2 r + \map f a} \paren {\paren {2 s + \map f b} \paren {2 t + \map f c} }\) Integer Multiplication is Associative
\(\ds \) \(=\) \(\ds a \times \paren {b \times c}\) Definition of Odd Integer and Definition of Even Integer

$\blacksquare$


Sources