Exercise $10\mathrm{.}1\mathrm{.}7$: $NP$ is also closed under each of the operations listed for $P$

in Exercise $10\mathrm{.}1\mathrm{.}6,$ with the $($presumed$)$ exception of $($f$)$ complementation. It is

not known whether or not $NP$ is closed under complementation, an issue we

discuss further in Section $11\mathrm{.}1.$ Prove that each of Exercise $10\mathrm{.}1\mathrm{.}6($a$)$ through

$($e$)$ holds for $NP.$