Let $\backslash$Sigma $=\{0,1\},$ L is a given regular language over $\backslash$Sigma $,$ wa in L$,$ and wr in $/$ L$.$ You are free to choose wa

and wr appropriately based on L such that $|$wa$|>=5$ and $|$wr$|>=5($you may need to choose shorter

strings if the language does not accept $/$ reject strings of length $>=5$ and you may need to choose longer

strings if the language does not accept $/$ reject strings of length State your choice.

$($a$)$ Design an NFA N for L$.$ Show the working of N for wa and wr$.$

$($b$)$ Design a right linear regular grammar G for L$.$ Show the derivation of wa and wr using G$.$

$($c$)$ Prove: L $=$ L$($N$)=$ L$($G$).$

Solve for the following languages:

Note: When we write $w=w_1{w}_{2}cdots{w}_n,$ we mean $w_{i}in$ and $|w|=n0($unless specified otherwise$).$

$L_{\text{I}}=\{w=w_1{w}_{2}cdots{w}_n|(wlon{)}^{{?}^{?}}(w_i=0,i-=0(mod2)),n>0\}$

$L_{II}=\{w=w_1{w}_{2}cdots{w}_n||$#${?}_1(w_1{w}_{2}cdots{w}_i)-$#${?}_0(w_1{w}_{2}cdots{w}_i)|1,1in\}$

$L_{\text{I}\text{I}\text{I}}=$

$\{w=w_1{w}_{2}cdots{w}_n|w_{i-1}{w}_i{w}_{i+1}={000}^{{?}^{?}}{w}_{1}cdots{w}_{i-2}lo{n}^{{?}^{?}}{w}_{i+2}cdots{w}_{n}lon,3in-2,n5\}$

$L_{IV}=\{w=w^1{w}^{2}cdots{w}^n|w^{i}in\{0,10,110,1110,1111\},n0\}$