$(5)(15$ marks$)$ For the following Boolean function

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

$[$Must show the intermediate conversion steps$]$

a$)$ Obtain the truth table of F$.$

b$)$ Draw the logic diagram, using the original Boolean expression.

c$)$ Use Boolean algebra to simplify the function to a minimum number of literals.

d$)$ Obtain the truth table of the function from the simplified expression and show that

it is the same as the one in part of Question $5($a$).$

e$)$ Draw the logic diagram from the simplified expression and compare the total

number of gates with the diagram of part Question $5($b$).$

$(6)(5$ marks$)$ Express the complement of the following functions in sum$-$of$-$minterms form:

$[$Must show the intermediate conversion steps$]$

a$)F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(2,4,7,10,12,14)$

b$)F(x,y,z)=$prod$(3,5,7)$

$(7)(5$ marks$)$ Convert each of the following to the other canonical form:

$[$Must show the intermediate conversion steps$]$

a$)F(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(1,3,5)$

b$)F(A,B,C,D)=(3,5,8,11)$