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