Question $2(20$ points$)$: $,$ Consider the knapsack set $K=\{\begin{array}{ccc}\text{x}\text{i}\text{n}& \{0,& 1{\}}^7\text{:}11x_1+6x_2+6x_3+\end{array}$

$\{$:$5x_4+5x_5+4x_6+x_{7}19\}.$ We want to derive valid inequalities for this set. Please answer

the following questions:

$($a$)$ For each of the following inequalities, identify whether or not they are valid knapsack covers,

and explain why.

$(1)x_4+x_5+x_{6}2$

$(2)x_1+x_2+x_{6}2$

$(3)x_2+x_3+x_6+x_{7}3$

$(4)x_2+x_4+x_5+x_{6}3$

$(5)x_1+x_3+x_4+x_{5}3$

$(6)x_2+x_3+x_4+x_5+x_{6}4$

$($b$)$ For all valid knapsack covers in Part $($a$),$ identify whether or not they are minimal.