CS 390 Solutions to Homework 3

Textbook pp. 76 - 79:

2.59 (c) Induction on n.
Basis: n = 0. tex2html_wrap_inline44 . tex2html_wrap_inline46 . Hence tex2html_wrap_inline48 . Also tex2html_wrap_inline50 .
Hence it holds for n = 0.
Induction: Assume tex2html_wrap_inline54 for an arbitrary k.
tex2html_wrap_inline58 because (xy)r = yrxr.
tex2html_wrap_inline60

2.39 (b) L is the set of strings having exactly one a and any number of bs before and after a.

2.40 (d) Basis: tex2html_wrap_inline94
Induction: If tex2html_wrap_inline96 , then tex2html_wrap_inline98 , tex2html_wrap_inline100 , tex2html_wrap_inline102 , and tex2html_wrap_inline104 .
Extremal: Nothing is in U unless it is obtained by the Basis and Induction.

(f) Basis: tex2html_wrap_inline108
Induction: If tex2html_wrap_inline110 , then tex2html_wrap_inline112 and tex2html_wrap_inline114 .
Extremal: Nothing is in W unless it is obtained by the Basis and Induction.