Exercise ${18}_{?}$ Closure properties of regular languages

$(7$ points$)$

$($a$)$ Let $=\{a,b\}$ and let the following deterministic finite automata $M_1=(Z_1,,{}_1,s_1,E_1)$ and

$M_2=(Z_2,,{}_2,t_1,E_2)$ be given:

$($i$)$ Construct by means of the cross product construction given in the lecture a deterministic

finite automaton $M_S,$ which accepts the language

$T(M_S)=T(M_1)T(M_2)$

$($Note: You only have to specify reachable states.$)$

$($ii$)$ Give a deterministic finite automaton $M_V,$ which accepts the language

$T(M_V)=T(M_1)T(M_2)$

$($Note: You can construct this automaton using an adapted version of the cross product

construction.$)$

$($b$)$ Describe for two arbitrary deterministic finite automata $M_A=(Z_A,,{}_A,s_A,E_A)$ and $M_B=$

$(Z_B,,{}_B,s_B,E_B)$ how to construct a deterministic finite automaton $M$ that accepts the lan$-$

guage

$T(M)=T(M_A)T(M_B)$

Briefly describe why your procedure is correct!