Problem $3$: Suppose the alphabet given is $\{1,0\}.$ For the languages that are given below

construct Deterministic Finite Automata DFA.

Note: $c{d}^2=cdd,$ and $(cd{)}^2=cdcd$

$L=$ all w strings from $\{1,0{\}}^{*}$ that end with $00$

$L=$ all w strings from $\{1,0{\}}^{*}$ that have $0$ remainder when the length of w is divided by $3$

$L=$ all strings starting with ${10}^2,$ followed with any string w from $\{1,0{\}}^{*}$ and ending with $1^3$

$L=$ all strings from $\{1,0{\}}^{*}$ that contains at least one $0$ and exactly two $1\text{'}$s

$L=$ all strings from $\{1,0{\}}^{*}$ except $11$ and $111$

$L=$ all strings from $\{1,0{\}}^{*},$ such that in every string the number of times substring $10$ and $01$

appear is the same.