$(24$ pts$)$ For any linear programming problem in the standard form and its dual problem, label each of the following statements as true, false or maybe and then justify your answer.

a$.$ The sum of the number of constraints and the number of variables $($before adding the slack$/$surplus variables$)$ is the same for both the primal and dual problems.

b$.$ At each iteration, the Simplex method simultaneously identifies a basic feasible solution for the primal problem and basic feasible solution for the dual problem such that their objective function values are the same.

c$.$ If the primal problem has an unbounded objective function, then the optimal value of the objective function for the dual problem must be zero.

d$.$ If a constraint of the primal problem is not binding at the optimal solution, then the corresponding dual variable is zero.

e$.$ Given a maximization primal problem, every dual simplex pivot increases the value of the dual problem$$s objective function.

f$.$ If the primal has multiple optimal solutions, then the optimal dual solution is degenerate.