$2(10\%)$ 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 you answer:

$($a$)$ The feasible region of the IP is a subset of the feasible region for the corresponding LP

relaxation problem. $(3\%)$

$($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. $(3\%)$

$($c$)$ If a non$-$integer 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. $(4\%)$