Problem $17($Homework $2$ Problem $5).$ Let $$ be a finite alphabet and let $L_0,L_1$sube${}^{**}$ be languages such that $L_0{L}_1=\frac{O}{?}.$ Think of $L_0$ and $L_1$ as the sets of "bad" and "good" strings respectively. We wish to design an automaton which is guaranteed to reject every string in $L_0$ and accept every string in $L_1.$ Our automaton is allowed to behave arbitrarily on strings that are neither in $L_0$ nor in $L_1.$ In other words, our automaton is required to recognize some language Lsube${}^{**}$ such that $L{L}_0=\frac{O}{?}$ and $Lsupe{L}_1.$

For concreteness, let us say that we are looking for a DFA over the alphabet $=\{0,1\}.$ Also, let $|)$:$\}$ the number of $0\text{'}$s and $1\text{'}$s in $x$ are unequal