# Define a binary operation $$\ast$$ on the set $$\{0, 1, 2, 3, 4, 5\}$$ as $a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a+b < 6} \\ a+b-6, & \quad \text{ if a+b \geq 6} \\ \end{array} \right.$ Zero is the identity for this operation. True or False?

Toolbox:
• An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
Given the set $X=\{0,1,2,3,4,5\}$ where the binary operation $\ast$ is defined by $a * b= \left\{ \begin{array}{1 1} a+b & \quad if\;a+b < 6\\ a+b-6 & \quad if a+b \geq 6 \end{array} \right.$
An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
To check if zero is the identity, we see that $a*0=a+0=a \qquad for\;a \in x$ and also $0*a=0+a=a \qquad for \;a \in x$
Given $a \in X, \qquad a+0 < 6\;$ and also $\;0+a < 6$
$\Rightarrow 0$ is the identify element for the given given operation