Consider the NFA in the picture

In the construction process while converting to a DFA, do not forget to account for the

lambda transitions.

For example, the first step is to identify the initial state, say $A$

$A=\{0\}$ is not correct as the lambda transitions enable the starting to be $1.$ You can make a

transition from $0$ to $1$ on lambda without scanning any symbol. Similarly, you can transition

to $2,3,7.$

So the starting state of the DFA $=\{0,1,2,4,7\}$

The next step is to define transitions from $A,$ say $A$ to $B$ on symbol a$.$

Check that $B=\{1,2,3,4,6,7,8\}$

$($i$)$ Give a complete equivalent description of the DFA.

$($ii$)$ Convert the above to a minimal DFA.