April 3, 1997
1 (a)
(b) 
({4}) = {1,2,3,4}.
(c) 
(1, aa) = {1,2,3,4}.
2.
{1,2,3,4} and {2} are the accepting states.
4. Basis: 
by the definition of 
.
Induction: Assume 
for an arbitrary k.
Then 
Now we are going to show 
(1) 
by the definition of 
.
(2) 
?
Basis: The basis of 
is . Hence
the basis of is a subset of .
Induction: Assume that for an arbitrary 
,
holds.
Then since , by the inductive clause of the definition of
, 
.
Hence holds. QED for (2).
Thus with (1) and (2) holds.
Hence 
.
This completes the induction and has been proven.