Prove Proposition $8\mathrm{.}3.$Proposition $8\mathrm{.}3$ Consider the mixed integer program $(8\mathrm{.}1)$ and its linear

programming relaxation $(8\mathrm{.}2).$ Then the following facts hold.

$($a$)$ The optimal value of the relaxation $(8\mathrm{.}2)$ is less than or equal to the optimal

value of the mixed integer linear program $(8\mathrm{.}1).$

$($b$)$ If the relaxation $(8\mathrm{.}2)$ is infeasible, then so is the mixed integer linear pro$-$

gram $(8\mathrm{.}1).$

$($c$)$ If the optimal solution $x^{**}$ of the relaxation $(8\mathrm{.}2)$ satisfies $x_j^{**}$inZ for jinJ

then $x^{**}$ is also an optimal solution to the mixed integer linear program $(8\mathrm{.}1).$