I need to prove $L_1L_3$ − $L_2$ where $L_1,L_3,L_2$ are regular languages. I proved via intersection way, but I need to show it with NFA. Because of complementary in intersection proof, I cannot prove it with NFA. Can anybody help me?
Maybe this is more understandable with all question. Question: Prove that if $L_1,L_3$ and $L_2$ are regular languages over the alphabet Σ = {a, b} , then so is
L = ($L_1L_3$−$L_2$) ∪ $L_4$ in which $L_4$ = {$w | w = bx(ab)^∗$, x ∈ L2} by using NFA.
You only know that $L_1$, $L_2$, and $L_3$ are regular. Hence, you can prove that $L_4$ is also regular, but you cannot draw automata that accept those languages unless you are given their definitions.
What you could do is describe a construction that, applied to automata for $L_1$, $L_2$, and $L_3$---whatever they may be---computes an automaton for $L_4$ and eventually an automaton for $(L_1L_3 - L_2) \cup L_4$.
The existence of such a construction is guaranteed by the closure properties of regular languages, and by Kleene's theorem, which says that the regular expression appearing in the definition of $L_4$ has an equivalent automaton.
However, describing such a construction is not necessary to prove the regularity of $(L_1L_3 - L_2) \cup L_4$. For instance, $L_1L_3$ is regular because regular languages are closed under concatenation.
I'm in no position to divine what your instructor or textbook authors meant by the words "by using NFA." A lot of the communication that takes place in a course relies on context and on a set of conventions that are specific to that course. Maybe the purpose of the exercise is to test your understanding of closure properties by having you rehash the proofs? The most qualified people to answer this question are your instructor and teaching assistants.
As for the doubt you express at the beginning of your post about complementation and NFAs, here's what I meant in my comment.
To complement a regular language given an NFA for it, we construct a DFA from the NFA and complement the DFA. (I assumed that you knew that.) Determinization may incur an exponential blow-up, but as long as we are merely proving the existence of an automaton, we don't care.
Besides, a DFA is also an NFA, which means that, strictly speaking, all steps of the construction of an automaton for $L_1L_3 - L_2$ from automata for $L_1$, $L_2$, and $L_3$ manipulate NFAs.
Bottom line: the need to complement one operand to show the regularity of the difference of two regular languages is not an issue, and you shouldn't worry about it.