Consider the following statements about any pure IP problem (in maximization form) and its LP relaxation. Label each of the statements as True or False, and then justify your answer.

(a) The feasible region for the LP relaxation is a subset of the feasible region for the IP problem.

(b) If an optimal solution for the LP relaxation is an integer solution, then the optimal value of the objective function is the same for both problems.

(c) If a noninteger solution is feasible for the LP relaxation, then the nearest integer solution (rounding each variable to the nearest integer) is a feasible solution for the IP problem.