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Math 239
Assignment 4
1. For each of the following sets
A
, determine whether
A
*
is uniquely created. Prove your
assertion in each case. (Hint: to prove a set
A
*
is uniquely created, try using induction
on the length.)
(a)
A
=
{
00
,
010
,
0111
}
(b)
A
=
{
010
,
0010
,
01001
}
.
Solution:
(a) In this case,
A
is uniquely created. We prove by induction on
n
that every
σ
∈
A
of length
n
is uniquely created.
If
n
= 0 then the only string in
A
*
of length 0 is the empty string, which is
uniquely created.
Suppose
n
≥
1 and that the statement holds for smaller values of
n
. Let
σ
=
a
1
...a
n
. So
a
1
= 0 and
n
≥
2. Consider
a
2
.
–
If
a
2
= 0 then
σ
= 00
a
3
n
where
a
3
n
∈
A
*
and by the induction
hypothesis
a
3
n
is uniquely created. Therefore
σ
is uniquely created.
–
If
a
2
= 1 then
σ
= 01
a
3
n
and
n
≥
3. Consider
a
3
:
*
If
a
3
= 0 then
σ
= 010
a
4
n
where
a
4
n
∈
A
*
and by the induc
tion hypothesis
a
4
n
is uniquely created. Therefore
σ
is uniquely
created.
*
If
a
3
= 1 then
σ
= 0111
a
5
n
where
a
5
n
∈
A
*
and by the induc
tion hypothesis
a
5
n
is uniquely created. Therefore
σ
is uniquely
created.
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 Spring '09
 M.PEI
 Sets, Recursion, ASCII, Structural induction, binary strings, A∗

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