Problem $3\mathrm{.}20.$ A safe has five locks, $v,w,x,y,$ and $z,$ all of which must be unlocked

for the safe to open. The keys to the locks are distributed among five executives in the

following manner:

A has keys for locks $v$ and $x$;

$B$ has keys for locks $v$ and $y$;

$C$ has keys for locks $w$ and $y$;

$D$ has keys for locks $x$ and $z$;

$E$ has keys for locks $v$ and $z.$

$($a$)$ Determine the minimum number of executives required to open the safe.

$($b$)$ Find all the combinations of executives that can open the safe. Write an expression

$f(A,B,C,D,E)$ which specifies when the safe can be opened as a function of

which executives are present.

$($c$)$ Who is the "essential executive" without whom the safe cannot be opened?