Draw logic diagrams of the circuits that implement the original and simplified expressions

in Problem $2\mathrm{.}2.$Simplify the following Boolean expressions to a minimum number of literals:

$($a$)*xy+x{y}^{\text{'}}$

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

$($c$)*xyz+x^{\text{'}}y+xy{z}^{\text{'}}$

$($d$)*(A+B{)}^{\text{'}}(A^{\text{'}}+B^{\text{'}}{)}^{\text{'}}$

$($e$)(a+b+c^{\text{'}})(a^{\text{'}}{b}^{\text{'}}+c)$

$($f$)a^{\text{'}}bc+ab{c}^{\text{'}}+abc+a^{\text{'}}b{c}^{\text{'}}$