Simplifying Boolean Equations: Simplify the following Boolean

equations. Show your work and list which axiom or theorem you used in each step. Your final

equation should be in minimized sum$-$of$-$products $($SOP$)$ form.

$($a$)$ W $=$ B$\text{'}($A $+$ BC $+$ B$\text{'}$CD $+$ A$\text{'}$D$)$

$($b$)$ X $=($A$+$B$\text{'})($A$\text{'}+$B$\text{'}+$D$)($B$\text{'}+$C$+$D$\text{'})$

$($c$)$ Y $=$ AB$\text{'}+($A$\text{'}$B$+($C$\text{'}+$D$\text{'})+$E$)$

$($d$)$ Z $=($A$+$B$\text{'}$C$)($AB $+$ BC $+$ A'BC' $+$ B$($D $+$ ABC$))$

$($e$)$ M $=($A $+$ B$\text{'}$C$)($A$\text{'}$D $+$ AB $+$ AB$$C $+$ AC$\text{'})$B