$3\mathrm{.}1*$ Simplify the following Boolean functions, using three$-$variable K$-$maps:

$F(x,y,z)=(0,2,4,5)$

$F(x,y,z)=(0,2,4,5,6)$

$F(x,y,z)=(0,1,2,3,5)$

$F(x,y,z)=(1,2,3,7)$

$3\mathrm{.}4$ Simplify the following Boolean functions, using K$-$maps:

$($a$)*F(x,y,z)=(2,3,6,7)$

$($b$)**F(A,B,C,D)=(4,6,7,15)$

$($c$)*F(A,B,C,D)=(3,7,11,13,14,15)$

$($d$)*F(w,x,y,z)=(2,3,12,13,14,15)$

$3\mathrm{.}6$ Simplify the following Boolean expressions, using four$-$variable K$-$maps:

$($a$)*A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}{D}^{\text{'}}+A{C}^{\text{'}}{D}^{\text{'}}+B^{\text{'}}C{D}^{\text{'}}+A^{\text{'}}BCD+B{C}^{\text{'}}D$

$($b$)**x^{\text{'}}z+w^{\text{'}}x{y}^{\text{'}}+w(x^{\text{'}}y+x{y}^{\text{'}})$

$($c$)A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}D+A{B}^{\text{'}}D+A^{\text{'}}B{C}^{\text{'}}+$ABCD$+A{B}^{\text{'}}C$

$($d$)A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}{D}^{\text{'}}+B{C}^{\text{'}}D+A^{\text{'}}{C}^{\text{'}}D+A^{\text{'}}BCD+AC{D}^{\text{'}}$

$3\mathrm{.}9$ Find all the prime implicants for the following Boolean functions, and determine which are essential:

$($a$)*F(w,x,y,z)=(0,2,4,5,6,7,8,10,13,15)$

$($b$)*F(A,B,C,D)=(0,2,3,5,7,8,10,11,14,15)$

$($c$)F(A,B,C,D)=(2,3,4,5,6,7,9,11,12,13)$

$($d$)F(w,x,y,z)=(1,3,6,7,8,9,12,13,14,15)$