We have:
and M be a finite automata.
Suppose d(M) be a deterministic automata that equivalence to M. if $M_1$ and $M_2$ be two finite automata, $M_1 + M_2$ is the finite automata such that the language of this automata is the union of language $M_1$ and $M_2$.
if $G_1$ and $G_2$ be two regular grammar such that language of these be $M_1$ and $M_2$, how the author conclude:
$L(G_1)-L(G_2)=L(\overline{d(\overline{d(M_1)}+M_2))} $
is true.
my question is why for example the following false:
$L(G_1)-L(G_2)=L(\overline{\overline{d(M_1)}+\overline{d(M_2)}})$
I ran into another question why before complement we should deterministic the automat?
This is a good example of how bad notation can make simple questions much more difficult.
(1) There is no reason to introduce grammars. You may as well denote by $L_1$ the language accepted by $M_1$ and by $L_2$ the language accepted by $M_2$.
(2) Let us denote by $L^c$ the complement of a language $L$. Then your first (correct) equality amounts to prove that $L_1 - L_2 = (L_1^c \cup L_2)^c$. Your second expression however would be $(L_1^c \cup L_2^c)^c = L_1 \cap L_2$.
As for your last question, you cannot directly compute an automaton accepting the complement of an automaton if you start with a nondeterministic automaton. Actually, there are known examples of a language with an $n$-state minimal DFA for which the minimal DFA of its complement has $2^n$ states.