$a$ 米 $b= \begin{cases}a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6\end{cases}$ Show that zero is the identity for this operation and each element $a \neq 0$ of the set is invertible with 6 - a being the inverse of $a$.

Solution:

Let $X=\{0,1,2,3,4,5\}$.

The operation * on $\mathrm{X}$ is defined as:

$a * b= \begin{cases}a+b & \text { if } a+b\lt6 \\\\ a+b-6 & \text { if } a+b \geq 6\end{cases}$

An element $e \in X$ is the identity element for the operation *, if $a * e=a=e * a …