$1\mathrm{.}16$ Use the construction given in Theorem $1\mathrm{.}39$ to convert the following two nonde$-$

terministic finite automata to equivalent deterministic finite automata.

$($a$)$

$($b$)$

Give regular expressions generating the languages. For $($a$),$ prove $($by induction$)$ that the language of your expression is equal to the language as described in $($a$).$