For each part, either prove that the statement is correct or find a counterexample. The primal

problem is denoted by $P,$ and the dual of it is denoted by $D.($Note that here the word prove does

not mean rigorous mathematical proofs.$)$

$($i$)$ If $P$ is infeasible, then $D$ must be unbounded.

$($ii$)$ If $P$ has an optimal solution, then both $P$ and $D$ must be feasible.

$($iii$)$ If $P$ does not have an optimal solution, it is either infeasible or unbounded.

$($iv$)$ If $P$ does not have an optimal solution, $D$ is either infeasible or unbounded.

$($v$)$ The optimal solution of $P$ must be smaller than the objective value of any feasible solution

of $D.$

$($vi$)$ Both $P$ and $D$ can be unbounded.

$($vii$)$ You will use Big$-$M method to find an initial solution to $P.$ After some simplex iterations, you

realize that the problem with artificial variables added is unbounded. Given this information,

only the following cases are possible:

$($a$)P$ is infeasible and $D$ is unbounded.

$($b$)P$ is infeasible and $D$ is infeasible.

$($c$)P$ is unbounded and $D$ is infeasible