Exercise $10\mathrm{.}1\mathrm{.}6$: As classes of languages, $P$ and $NP$ each have certain closure

properties. Show that $P$ is closed under each of the following operations:

a$)$ Reversal.

b$)$ Union.

$*!$ c$)$ Concatenation.

$!$ d$)$ Closure $($star$).$

e$)$ Inver se homomorphism.

f$)$ Complementation.