$($b$)$ Design and implement a digital circuit using minimum number of $2-$

input NAND gates only. The circuit has four inputs $(w,x,y$ and $z)$

and three outputs $($F$1,$ F$2$ and F$3).$ F$1,$ F$2$ and F$3$ outputs are

represented by following Boolean expressions.

$F1(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(0,2,4,6,7,9,11,13)$

$F2(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(0,1,4,6,8,10,14)$

$F3(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(1,3,4,7,9,12,15)$

Show your complete work based on K$-$map based simplification and

your overall circuit design must employ minimum number of $2-$input

NAND gates.

$(12$ marks$)$

$($c$)$ Implement the Boolean function, $F2(w,x,y,z)$ in part $($b$)$ by using a

$3-$to$-8$ decoder $($given in Appendix$)$ along with minimum number of

gates.

$(7$ marks$)$