CSCI 2400  Models of Computation
Homework 1 Solutions
Problem 1 (25 points)
Prove that if
A
and
B
are languages over the same alphabet Σ and
A
⊆
B
, then
A
*
⊆
B
*
.
Solution:
We will prove by mathematical induction that
A
k
⊆
B
k
, for all
k
≥
0.
Base case:
A
0
⊆
B
0
⇒ {
λ
} ⊆ {
λ
}
√
Inductive hypothesis: Assume that
A
k
⊆
B
k
for some k
≥
0.
√
Inductive step: Given that
A
k
⊆
B
k
, demonstrate that
A
k
+1
⊆
B
k
+1
:
A
k
⊆
B
k
A
k
·
A
⊆
B
k
·
A
⊆
B
k
·
B
A
k
+1
⊆
B
k
+1
Thus, we can show that
A
k
⊆
B
k
for any positive, Fnite k by building up
from our base case, one step at a time. It follows that because
A
*
=
A
0
∪
A
1
∪
. . .
and
B
*
=
B
0
∪
B
1
∪
. . .
, then
A
*
⊆
B
*
.
Problem 2 (25 points)
Prove that there does not exist a language
L
such that:
L
*
=
{
a
}
*
{
b
}
*
.
Solution:
Assume that there is such a language, and attempt to Fnd a contra
diction.
{
a
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 CAROTHERS
 Logic, Inductive Reasoning, Natural number, Mathematical logic, ak bk

Click to edit the document details