Consider the following functions

d $=$ a b c$+$ b$$ c

e $=$ a$$ b$$ c

f $=$ c$($b$+$ a$)+$ a$$ b$$

For what follows, define $$cost$$ as the number of literals in the expression. $($A $$literal$$

is an instance of a variable, whether negated or not.$)$ In each case, provide the lowest

cost expression. Include the given function in an expression for the target function

only if this reduces the cost; otherwise, just say that the lowest cost expression is in

terms of the input variables only.

$($a$)$ Express d in terms of f$,$ as well as the input variables. $[4$ points$]$

$($b$)$ Express e in terms of d$,$ as well as the input variables. $[4$ points$]$

$($c$)$ Express f in terms of e$,$ as well as the input variables.