$[2][20$ 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 and show used Theorems and Axioms$).$

$($a$)F(a,b,c)=a^{\text{'}}*b^{\text{'}}*c^{\text{'}}+a*c^{\text{'}}+a^{\text{'}}*b^{\text{'}}*c+a*c$

$($b$)F(x,Y)=(x^{\text{'}}+Y)*(x^{\text{'}}+Y^{\text{'}})+1$

$($c$)$

z$)$

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