Exercise $3\mathrm{.}6([$aux$16]$ p$16)$

Prove that the following are equivalent:

$($a$)V_1$ is a vertex cover of $G.$

$($b$)V-V_1$ is an independent set in $G.$

and, continuing, prove that the following are equivalent:

$($a$)V_2$ is an independent set in $G.$

$($b$)V_2$ is a clique in $G^c.$