91Ó°ÊÓ

Given language is$A/B=\left\{\mathrm{Ó\neg }|\mathrm{Ó\neg } \mathrm{χ} \mathrm{A}\mathrm{for}\mathrm{some}\mathrm{χ} ∈\mathrm{B}\right\}$

Here, A is a regular language and B is any language.

Since A is a regular language, some DFA will be recognize the language A .

Let$M=(Q,∑,\mathrm{δ},q_{0}F)$be that which recognizes.

Here, Q is the set of states.

$∑$is set of alphabets = of the alphabets for A and B.

$\mathrm{δ}$ is the transition function.

$q_0$is the start state.

F is the set of final states.

To prove $A/B$ is a regular language, construct a DFA, that recognizes the language$A/B$

Let $M\text{'}=(Q\text{'},∑\text{'},{\mathrm{δ}}_0^{\text{'}},F\text{'})$be the DFA, that recognizes $A/B$.

$Q\text{'}$= set of states =role="math" localid="1663154225010" $Q$

$∑$=set of alphabets =$∑$

$\mathrm{δ}\text{'}$= transition function =$\mathrm{δ}$

$q_0^{\text{'}}$=start state=$q_0^{}$

$F\text{'}=\{q∈Q|∃\mathrm{χ}∈B$, such that M goes from q to some sate in F on reading $\mathrm{χ}$}

Thus, a DFA, $\mathrm{M}\text{'}$to recognize the language $A/B$ has been constructed.

Hence, $A/B$is a regular language.