. Then since 
and
, 
.
Hence 
. Inductive Step: Assume that
and .
Then 
and
. Hence 
.
June 17, 1998
1 (a) Basis Step: Let . Then since and
, .
Hence .
Inductive Step: Assume that and .
Then and
. Hence .
(b) Let 
, and
. Then 
.
But 
.
Hence 
does not hold in general.
2. Let L denote the given language. Also let 
.
Then any pair of distinct strings 
and 
of S can be distinguished
with respect to L by
using 
. Hence L is not regular.
3.
4 (a) 
, R(3,2,1) = aa + b,
, and R(2,3,1) = a.
Hence 
Hence 
