$($please ans it ASAP$)$ Consider the DFAs $M_1($on the left$)$ and $M_2($on the right$)$:

$($a$)$ If we use the general constructions discussed in class and in the book for building a DFA whose

language is

$L(M_1)L(M_2)$

how many states would be in this DFA? Briefly justify your calculation.

$($b$)$ Draw the state diagram that results from this construction and remove any unreachable states.

How many states are left?

$($c$)$ Describe the language $L(M_1)L(M_2)$ in set builder notation or as a regular expression.

$($d$)$ Is there any way to build a machine that recognizes the same language $(L(M_1)L(M_2))$ but uses

fewer states? $($You can either draw the result or describe in words how to construct it$.)$