1. Use the construction given in the proof of Theorem 1.45 to give the state diagram of an NFA recognizing the union of the two languages recognized by M1 and M2: (L(M1) ∪ L(M2)).

2. Use the construction given in the proof of Theorem 1.47 to give the state diagram of an NFA recognizing the concatenation of the two languages recognized by M1 and M2: (L(M1) • L(M2)).

3. Use the construction given in the proof of Theorem 1.49 to give the state diagram of an NFA recognizing the star of the language recognized by DFA M1: (L(M1)*).

4. (Exercise 1.19b) Use the procedure described in Lemma 1.55 to convert the following regular expression to a nondeterministic finite automaton.

(((00)^*(11)) U 01 )^*

5. (Exercise 1.21b) Use the procedure described in Lemma 1.60 to convert the following finite automaton to a regular expression. Remove states in the following order: 1, 2, 3.

6. (Exercise 1.30) Describe the error in the following "proof" that 0^*1^* is not a regular language. (An error must exist because 0^*1^* is regular.) The proof is by contradiction. Assume that 0^*1^* is regular. Let p be the pumping length for 0^*1^* given by the pumping lemma. Choose s to be the string 0^p1^p. You know that s is a member of 0^*1^*, but Example 1.73 shows that s cannot be pumped. Thus you have a contradiction. So 0^*1^* is not regular.

7. If we try to apply the pumping lemma to a regular language, we cannot complete the proof. Explain what would go wrong if we tried to apply the pumping lemma to the language L = { } (L is the empty set).

8. Use the pumping lemma to show that the following languages are not regular:

a. { bⁿ aⁿ⁺³ | n ≥ 0}
b. { a^{2 n} | n ≥ 0}

9. For each of the following languages, determine if the language is regular and prove your answer:

a. { w | w has an odd number of a's}
b. { w | w has an odd number of a's and even number of b's}
c. { w | w has two more a's than b's}