$7-$ For the Boolean function $(12$ points$)$

F $=$ xy$$z $+$ x$$y$$z $+$ w$$xy $+$ wx$$y $+$ wxy

$($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 $($a$).$

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

of gates with the diagram of part $($b$).$