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