Problem $3$: Suppose the alphabet given is $\{a,b\}.$ For the languages that are given below

construct Deterministic Finite Automata DFA.

$L=$ all $w$ strings from $\{a,b{\}}^{**}$ that end with $bb$

$L=$ all $w$ strings from $\{a,b{\}}^{**}$ that have $0$ remainder when the length of $w$ is divided by $3$

$L=$ all strings starting with $a{b}^2,$ followed with any string $w$ from $\{a,b{\}}^{**}$ and ending with $a^3$

$L=$ all strings from $\{a,b{\}}^{**}$ that contains at least one $b$ and exactly two $a\text{'}$s

$L=$ all strings from $\{a,b{\}}^{**}$ except aa and aaa

$L=$ all strings from $\{a,b{\}}^{**},$ such that in every string the number of times substring $ab$ and $ba$

appear is the same.