$(4)(10$ marks$)$ Simplify the following Boolean expressions, using four$-$variable k$-$maps:

$[$Must mark the calculating squares inside the k$-$map$]$

a$)\frac{?}{\text{b}\text{a}\text{r}}(w)*z+x*z+\stackrel{}{x}*y+w*\stackrel{}{x}*z$

b$)A\frac{*}{b}ar(D)\frac{+}{b}ar(B)\frac{*}{b}ar(C)*D+B*C\frac{*}{b}ar(D)+B\frac{*}{b}ar(C)*D$

$(5)(10$ marks$)$ Simplify the following expressions to $($i$)$ sum$-$of$-$products and $($ii$)$ products$-$of$-$sums: $[$Must mark the calculating squares inside the k$-$map$]$

$A*C\frac{*}{b}ar(D)\frac{+}{b}ar(C)*D+A*B*C*D$

$(6)(10$ marks$)$ Simplify the following Boolean function F$,$ together with the don't$-$care conditions d$,$ and then express the simplified function in sum$-$of$-$min terms form:

a$)$

$F(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(0,1,4,5,6)$

$d(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(2,3,7)$

b$)$

$F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(5,6,7,12,14,15)$

$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(3,9,11)$

$(7)(10$ marks$)$ Simplify the following functions, and implement them with two$-$level NOR gate circuits:

$F=w*\stackrel{}{x}\frac{+}{b}ar(y)\frac{*}{b}ar(z)\frac{+}{b}ar(w)*y\frac{*}{b}ar(z)$