$20$ points In the videos and the text, you saw that regular languages

are closed under the union operation $$ i$.$e$.,$ if L$1$ and L$2$ are both

regular, then L$1\backslash$cup L$2$ is regular. There were $2$ proofs. In both proofs,

we built a machine from $2$ DFA$$s M$1$ and M$2,$ where L$($M$1)=$ L$1$ and

L$($M$2)=$ L$2.$ The first proof constructed a new DFA and the second

constructed and NFA. $($Recall that if and only if proofs have $2$ parts $$

the $$if$$ part and the $$only if$$ part.$)$

Describe how to modify the DFA construction to prove that regular

languages are closed under intersection. Argue that your modified con$-$

struction is correct. Specifically, prove that w in L$1\backslash$cap L$2$ if and only if

w is accepted by your constructed DFA