$[5][20$ points total, $5$ points each$]$ Simplification of Logic Functions using Boolean

Algebra

Simplify the following Boolean expressions to a minimum number of literals using only

Boolean algebra $($show all your steps for credit$).$

$($a$)$ F$($x$,$ y$,$ z$)=$ x$\text{'}$ y$\text{'}$ z$\text{'}+$ x $$ z $+$ x$\text{'}$ y$\text{'}$ z

$($b$)$ F$($X$,$ Y $)=($X$\text{'}+$ Y $)($X$\text{'}+$ Y$\text{'})$

$($c$)$ F$($x$,$ y$,$ z$)=($x $+$ y $+$ z$\text{'})($x$\text{'}+$ y $+$ z$\text{'})($x $+$ y $+$ z$)($x$\text{'}+$ y $+$ z$)$

$($d$)$ F$($x$,$ y$,$ z$)=$ x$\text{'}$ y$\text{'}$ z $+$ x $$ y $$ z$\text{'}+$ x$\text{'}$ y $$ z $+$ x $$ y$\text{'}$ z$\text{'}$