vUsing the images answer all of these questions: Note: DFA is deterministic, NFA is

non$-$deterministic, FSA can be either.

$1.$ Closure Properties:

a$)$ Prove, using DFAs, that the class of regular languages is closed under the intersection

operation. Note: This should be a formal proof. See the proof for union in the textbook

for ideas.

b$)$ Given the DFA M$1$ that recognizes L$1$ and M$2$ that recognizes L$2,$ both below;

i$.$ Write a $5-$tuple defining the DFA M which recognizes the intersection of L$1$ and L$2$

by running M$1$ and M$2.$ Thoroughly define each element of the tuple.

ii$.$ Draw the state diagram for M$.$ The label for each state should be a pair of states.

iii. Give the computation in M for the string $10101.($See the definition of

computation from notes or the text.$)$

iv$.$ Is $10101$ accepted by M$?$ Is $10101$ an element of L$1\backslash$cap L$2?$ Explain.

c$)$ Regular Languages are closed under set difference where set difference of two

languages L$1$ and L$2$ is defined to be the elements in L$1$ that are not in

L$2.$

i$.$ Write set difference symbolically using the other known closure properties.

Explain how that proves that set difference is closed.

ii$.$ Write a $5-$tuple defining the DFA M$$ which recognizes the set difference of L$1$ and

L$2.$ Thoroughly define each element of the tuple.

iii. Draw the state diagram for M$.$ The label for each state should be a pair of

states