# Ring of Integers Modulo 2/Cayley Tables

## Cayley Tables for Ring of Integers Modulo $2$

$\struct {\Z_2, +_2, \times_2}$

can be described completely by showing its Cayley tables:

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

They can be presented more simply as:

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