# Isomorphism between Ring of Integers Modulo 2 and Parity Ring

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## Theorem

The ring of integers modulo $2$ and the parity ring are isomorphic.

## Proof

To simplify the notation, let the elements of $\Z_2$ be identified as $0$ for $\eqclass 0 2$ and $1$ for $\eqclass 1 2$.

Let $f$ be the mapping from the parity ring $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ and the ring of integers modulo $2$ $\struct {\Z_2, +_2, \times_2}$:

$f: \struct {\set {\text{even}, \text{odd} }, +, \times} \to \struct {\Z_2, +_2, \times_2}$:
$\forall x \in R: \map f x = \begin{cases} 0 & : x = \text{even} \\ 1 & : x = \text{odd} \end{cases}$

The bijective nature of $f$ is apparent:

$f^{-1}: \struct {\Z_2, +_2, \times_2} \to \struct {\set {\text{even}, \text{odd} }, +, \times}$:
$\forall x \in \Z_2: \map {f^{-1} } x = \begin{cases} \text{even} & : x = 0 \\ \text{odd} & : x = 1 \end{cases}$

Thus the following equations can be checked:

 $\ds 0 +_2 0 = \ \$ $\ds \map f {\text{even} } +_2 \map f {\text{even} }$ $=$ $\ds \map f {\text{even} + \text{even} }$ $\ds = 0$ $\ds 0 +_2 1 = \ \$ $\ds \map f {\text{even} } +_2 \map f {\text{odd} }$ $=$ $\ds \map f {\text{even} + \text{odd} }$ $\ds = 1$ $\ds 1 +_2 0 = \ \$ $\ds \map f {\text{odd} } +_2 \map f {\text{even} }$ $=$ $\ds \map f {\text{odd} + \text{even} }$ $\ds = 1$ $\ds 1 +_2 1 = \ \$ $\ds \map f {\text{odd} } +_2 \map f {\text{odd} }$ $=$ $\ds \map f {\text{odd} + \text{odd} }$ $\ds = 0$

and:

 $\ds 0 \times_2 0 = \ \$ $\ds \map f {\text{even} } \times_2 \map f {\text{even} }$ $=$ $\ds \map f {\text{even} \times \text{even} }$ $\ds = 0$ $\ds 0 \times_2 1 = \ \$ $\ds \map f {\text{even} } \times_2 \map f {\text{odd} }$ $=$ $\ds \map f {\text{even} \times \text{odd} }$ $\ds = 0$ $\ds 1 \times_2 0 = \ \$ $\ds \map f {\text{odd} } \times_2 \map f {\text{even} }$ $=$ $\ds \map f {\text{odd} \times \text{even} }$ $\ds = 0$ $\ds 1 \times_2 1 = \ \$ $\ds \map f {\text{odd} } \times_2 \map f {\text{odd} }$ $=$ $\ds \map f {\text{odd} \times \text{odd} }$ $\ds = 1$

$\blacksquare$

These results can be determined from their Cayley tables:

### Cayley Tables for Parity Ring

$\begin{array}{r|rr} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array} \qquad \begin{array}{r|rr} \times & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{even} \\ \text{odd} & \text{even} & \text{odd} \\ \end{array}$

### Cayley Tables for $\Z_2$

$\begin{array} {r|rr} \struct {\Z_2, +_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array} \qquad \begin{array}{r|rr} \struct {\Z_2, \times_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 0 2 \\ \eqclass 1 2 & \eqclass 0 2 & \eqclass 1 2 \\ \end{array}$

They can be presented more simply as:

$\begin{array}{r|rr} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{r|rr} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$