$4.$ Consider the following flow network. $($The values are edge capacities.$)$

v$18//$

$1$

v$2$

$2$

&

&

$8$

s

$8$

$8$

$8$

$4//$

$3$

&

&

v$33//$

$7$

$2$

$8$

$8$

v$44//$ t

v$510$

$//$

$4$

$8$

$8$

v$6$

$2$

O

O $9$

$8$

$8$

$($a$)$ What is the value of the capacity of the cut $($S$,$ T $)$ where S $=\{$s$,$ v$1,$ v$2,$ v$4\}$ and T $=\{$v$3,$ v$5,$ v$6,$ t$\}?$

$($b$)$ Prove that the cut $($S$,$ T $)$ where S $=\{$s$,$ v$1,$ v$2,$ v$4\}$ and T $=\{$v$3,$ v$5,$ v$6,$ t$\}$ is a minimum cut? $($Hint:

Use the max$-$flow min cut theorem.