(a) (L*L)+ = L+ .
(b)
. (c)
. (d)
is regular if 
, 
, and 
are regular. (e) Suppose
, and 
is a function. Then there is
a function 
such that f(g(y)) = y for every y in B.
June 3, 1998
1. Which of the following statements are true and which are false ?
(a) (L*L)+ = L+
.
(b) .
(c) .
(d) is regular if , , and are regular.
(e) Suppose , and is a function. Then there is
a function such that f(g(y)) = y for every y in B.
2. Find a regular expression for each of the following languages:
(a) L defined recursively as 
; if 
, then 
and 
.
(b) The set of strings of odd length over {a,b}.
3. Prove by structural induction on string x that 
, where
is defined as
Basis Clause: 
Inductive Clause: If 
. then 
for all 
.
Extremal Clause: As usual.
Also for any string x, 
is defined as
Basis clause: 
Inductive Clause: If and , then
4. Find a regular expression for the language of strings over the alphabet {a,b} which are accepted by the FA given by the following transition table.
The initial state is 
and the accepting states are and 
.