Consider the following functions

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

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

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

$($a$)$ Express $d$ in terms of $e,$ as well as the input variables. $[3$ points$]$

$($b$)$ Express $d$ in terms of $f,$ as well as the input variables. $[3$ points$]$

$($c$)$ Express e in terms of $d,$ as well as the input variables. $[3$ points$]$

$($d$)$ Express $e$ in terms of $f,$ as well as the input variables. $[3$ points$]$

$($e$)$ Express $f$ in terms of $d,$ as well as the input variables. $[3$ points$]$

$($f$)$ Express $f$ in terms of $e,$ as well as the input variables. $[3$ points$]$

$($g$)$ Define "cost" as the number of literals in the expression. $($A "literal" is an

instance of a variable, whether negated or not.$)$ Assemble the lowest cost cir$-$

cuit, ensuring that it is combinational $($so all cycles are broken for each input

combination$).[7$ points$]$