Minimize each function using K$-$maps, stating the result in SOP $($sum of products$)$ form. Place

the variables on the K$-$map axes in alphabetical order beginning with the vertical axis. $($Recall the

minterm notation $F(a,b,c)={\displaystyle \underset{?}{\overset{?}{}}}m(1,2,3)$ is shorthand for $F(a,b,c)=\frac{}{b}ar(b)c+\frac{b}{b}ar(c)+bc.)$

$($a$)S(a,b,c)={\displaystyle \underset{?}{\overset{?}{}}}m(0,1,2,4,5,6)$

$($b$)P(A,B,C)=(A+B\frac{+}{b}ar(C))(A\frac{+}{b}ar(B)\frac{+}{b}ar(C))(A+B+C)(\frac{?}{\text{b}\text{a}\text{r}}(A)+B\frac{+}{b}ar(C))(\frac{?}{\text{b}\text{a}\text{r}}(A)\frac{+}{b}ar(B)\frac{+}{b}ar(C))$

$($Hint: $P$ is zero whenever any one of the factors in parentheses is zero, so to save algebra, put

zeros in the right places, then fill in the rest with ones.$)$

$($c$)R(u,v,w,x)={\displaystyle \underset{?}{\overset{?}{}}}m(0,2,3,7,8,9,10,11,13)$

$($d$)T(W,x,Y,Z)=YZ\frac{+}{b}ar(W)\stackrel{}{x}Y+W\frac{x}{b}ar\frac{Y}{b}ar(Z)+\frac{\stackrel{}{x}}{b}ar(Y)Z\frac{+}{b}ar\frac{W}{b}ar(Y)Z+Wx\frac{Y}{b}ar(Z)+\frac{x}{b}ar\frac{Y}{b}ar(Z)$