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### Chapter 11

#### Section 11.1

1. a. {a, b}.

c. {a, Λ, b, bb, . . ., bn, . . . }.

e. {a, b, ab, bc, abb, bcc, ..., abn, bcn, . . . }.

2. a. a + b + c. c. ab* + ba*. e. Λ + a(bb)*. g. Λ + c*a + bc*. i. a*bc*.

3. 0 + 1(0 + 1)*.

4. a. (aa + ab + ba + bb)*. c. (a + b)*aba(a + b)*.

5. a. (ab)*. c. a (a + b)*.

6. a.

$\begin{array}{ll}b+a{b}^{*}+a{a}^{*}b+a{a}^{*}a{b}^{*}& =b+a{b}^{*}+a{a}^{*}\left(b+a{b}^{*}\right)\\ =\left(\mathrm{\Lambda }+a{a}^{*}\right)\left(b+a{b}^{*}\right)\\ ={a}^{*}\left(b+a{b}^{*}\right)\phantom{0000000000}\left(\mathrm{by}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(11.1.1\mathrm{e}\right)\end{array}$

c. By using property 7 of (11.1.1g) the subexpression (a + bb*a)* of the left side am be written (a*bb*a)*a*. So the left expression has the following form:

ab*a(a + bb*a)*b = ab*a(a*bb*a)*a*b.

Similarly, the subexpression (b + aa*b)* of the right side of the original equation can be written as b*(aa*bb*)*. So the right expression has the following form:

a(b + ...

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