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Undergraduate Course
ELEMENTS OF COMPUTATION THEORY College of Computer Science
Chapter 1 Zhejiang University
Fall-Winter, 2007
P 46
1.7.4 Show each of the following.
(c) If a and b are distinct symbols, then { a, b } * = { a } * ( { b }{ a } * ) * .
(d) If Σ is an alphabet, e ∈ L
1
⊆ Σ * and e ∈ L
2
⊆ Σ * , then ( L
1
Σ * L
2
) = Σ * .
Solution:
(c) It is obvious that { a }
*
( { b }{ a }
*
)
*
⊆ { a, b }
*
.
On the other hand, suppose that w ∈ { a, b } * , then w = a * or w = a * ba * ba * · · · ba *
∈ { a }
*
( { b }{ a }
*
)
*
.
(d) Suppose w ∈ ( L
1
Σ * L
2
). Then w = xyz , where x ∈ L
1
⊆ Σ * , y ∈ Σ * , z ∈ L
2
⊆ Σ * .
Thus xyz ∈ (Σ
*
)
*
= Σ
*
.
On the other hand, suppose w ∈ Σ * . Then because e ∈ L
1
and e ∈ L
2
,
w = ewe ∈ ( L
1
Σ * L
2
).
1.7.6 Under what circumstances is L + = L * - { e } ?
Solution: L + = L * - { e } exactly when e 6∈ L .
P 51
1.8.3 Let Σ = { a, b } . Write regular expressions for the following sets:
(c) All strings in Σ
*
with exactly one occurrence of the substring aaa .
Solution: (( a ∪ aa ∪ b * ) b ) * aaa ( bb * ( a ∪ aa ∪ b * )) * .

1.8.5 Which of the following are true? Explain.
(a) baa ∈ a
*
b
*
a
*
b
*
(b) b * a * ∩ a * b * = a * ∪ b *
(c) a * b * ∩ b * a * = ∅
(d) abcd ∈ ( a ( cd )
*
b )
*
Solution:
(a) true. baaa consists of zero repetitions of a , followed by one repetition of