Show that the family of regular languages is closed under the left quotientwith a regular language

To show that the family of regular languages is closed under the left quotient with a regular language, we need to show that if L1 and L2 are regular languages, then L1\L2 is also a regular language. Here are the steps:

1. Let L1 and L2 be regular languages. By definition, there exist finite automata M1 and M2 that recognize L1 and L2 respectively.

2. Construct a new automaton M that recognizes L1\L2 as follows:

   a. The states of M are the same as the states of M1.

   b. The start state of M is the start state of M1.

   c. The accept states of M are those states q of M1 for which there exists a string w in L2 such that the transition function of M1 maps q and w to an accept state of M1.

   d. The transition function of M is the same as the transition function of M1.

3. We claim that M recognizes L1\L2. To see this, note that a string x is in L1\L2 if and only if there exists a string w in L2 such that xw is in L1. But by the construction of M, x is accepted by M if and only if there exists a string w in L2 such that the transition function of M1 maps the start state of M and xw to an accept state of M1. Hence, M recognizes L1\L2.

4. Since we have constructed a finite automaton that recognizes L1\L2, it follows that L1\L2 is a regular language.

Therefore, the family of regular languages is closed under the left quotient with a regular language.