Let $=\{0,1\},L$ is a given regular language over $,w_a$inL, and $w_r!$inL. You are free to choose $w_a$ and $w_r$ appropriately based on $L$ such that $|w_a|5$ and $|w_r|5($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 $5).$ State your choice.

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

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

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

Solve for the following languages:

$]$

$(1+(1+1)+(1+1))+$

$(2+(2+1)+(1+1))+$

$(3+(2+1)+(1+1))=[25$

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

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

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

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

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