November 22, 1995

		cs330 - Discrete Structures
		===========================

Homework #5 - Solutions
=======================

1
=
a) LM = {bba, ba, b, abbbba, abbba, abbb, bbba, bba, bb}
b) ML = {bba, bbaabb, bbab, ba, baabb, bab, b, babb, bb}
c) L^0 = {E}
   L^1 = L(L^0) = {E, abb, b}{E} = {E, abb, b}
   L^2 = L(L^1) = {E, abb, b}{E, abb, b} 
                = {E, abb, b, abbabb, abbb, babb, bb}

d) M^0 = {E}
   M^1 = {bba, ba, b}
   M^2 = {bba, ba, b}{bba, ba, b} =
       = {bbabba, bbaba, bbab, babba, baba, bab, bbba, bba, bb}

2
=
a) L = {b, ba}
b) L = {E, ba}
c) L = {E, b, ab}

3
=
a) L(a+b) = L(a)UL(b) = {a}U{b} = {a, b}
b) L(a+bc) = L(a)UL(bc) = L(a)U(L(b)L(c)) = {a}U({b}{c}) =
           = {a}U{bc} = {a, bc}

c) L(a+b*) = L(a)UL(b*) = L(a)UL(b)* = 
	   = {a}U{b}* = {a}U{E, b, bb, bbb, bbbb, .., b^n, ...}
	   = {E, a, b, bb, bbb, ...}

d) L = {c, a, ab, abb, abbb, ..., ab^n, ..}
e) L = {a, ab, abb, abbb, .., ab^n, .., b, bc, bcc, ..., bc^m, ..}
	m, n >= 0, natural numbers

f) L = {a^mbc^n | m, n >=0 } U {ac}
 In plain English: either the string ac or strings starting with any
		   number of a, followed by a b, followed by any number
		   of c.

4
=
a) a+b+c
b) a(aa)*
c) (a*)b(c*)
d) a* + b* + c*


5
=
0 + 1(0+1)*

6
=
a) ((a+b)(a+b))*
which is the same as
   (aa+ab+ba+bb)*

b) (a+b)*aba(a+b)*
c) see 4b