For the following Boolean function

$F=x{y}^{\text{'}}z+x^{\text{'}}{y}^{\text{'}}z+w^{\text{'}}xy+w{x}^{\text{'}}y+wxy$

$($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$)$