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 the five executives in the following manner:

Mr$.$ A has keys for locks v and x

Mr$.$ B has keys for locks v and y

Mr$.$ C has keys for locks w and y

Mr$.$ D has keys for locks x and z

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

$($i$)$ Determine the minimal number of executives required to open the safe.

$($ii$)$ Find all the combinations of executives that can open the safe.

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

Design the combinational circuit