Let $\backslash$Sigma be a finite alphabet and let L$0,$ L$1\backslash$Sigma

$$ be languages such that L$0\backslash$cap L$1=.$ 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 L $\backslash$Sigma

$$

such that L $\backslash$cap L$0=$ and L $$ L$1.$

For concreteness, let us say that we are looking for a DFA over the alphabet $\backslash$Sigma $=\{0,1\}.$ Also, let

L$0=\{$x in $\{0,1\}$

$$

$|$ the number of $0$s and $1$s in x are unequal$\}$ and L$1=\{0$

n$1$

n $|$ n in N $\backslash$cup $\{0\}\}.$

Observe that L$0\backslash$cap L$1=.$ Does there exist such a DFA which accepts every string in L$1$ and rejects

every string in L$0?$ Prove your answer.