Final answer:
The answers to the true or false questions are explained with reasoning. The first question is true because it is impossible for both the primal and the dual to be unbounded. The second question is false because a unique optimal solution in the primal does not guarantee a unique optimal solution in the dual. The third question is true because the objective value of a primal feasible solution will always be less than or equal to the objective value of a dual feasible solution when the primal is a minimization problem and the dual is a maximization problem.
Step-by-step explanation:
1. True. It is impossible for the primal and the dual to both be unbounded. This is because if the primal is unbounded, it means that the objective function can be increased indefinitely. In this case, the dual will be infeasible or have no feasible solution. Similarly, if the dual is unbounded, it means that the objective function can be decreased indefinitely, and the primal will be infeasible or have no feasible solution.
2. False. If the primal has a unique optimal solution, it does not guarantee that the dual will also have a unique optimal solution. The duality theorem states that if the primal has an optimal solution, then the dual must also have an optimal solution, but it does not guarantee uniqueness.
3. True. If both the primal and dual linear programs are feasible, and the primal is a minimization problem while the dual is a maximization problem, then the objective value given by any primal feasible solution will always be less than or equal to the objective value given by any dual feasible solution. This is a consequence of the duality theorem.