Let $=\{a,b\}.$ Consider the following regular grammars over $.5+(4^{**}1)+3+4+4=20$

$L=\{w=w_1{w}_{2}cdots{w}_n|(w_1=w_n^{{?}^{?}}|w|-=0(mod2))vv(w_1{w}_n^{{?}^{?}}|w|-=1(mod2)),n>0\}$

$1$

$L_s=\{w=w_1{w}_{2}cdots{w}_n|w_1=w_n,n>0\}$

$L_d=\{w=w_1{w}_{2}cdots{w}_n|w_1{w}_{n}n>0\}$

$L_e=\{w=w_1{w}_{2}cdots{w}_n||w|0(mod2),n>0\}$

$L_o=\{w=w_1{w}_{2}cdots{w}_n||w|1(mod2),n>0\}$

$($a$)$ Design a DFA $M$ for $L.$

$($b$)$ Design DFAs $M_s,M_d,M_e,M_o$ for $L_s,L_d,L_e,L_o$ respectively.

$($c$)$ Express $L$ in terms of $L_s,L_d,L_e,L_o.$

$($d$)$ Using the relationship in Q $(3$c$),$ connect the DFAs from Q $(3$b$)$ to get an NFA $N$ for $L.$

$($e$)$ Prove that $L(M)=L(N)=L.$