COT3100
Summer’2001
Recitation on Languages and Machines
07/19/2001
1.
Let
A
,
B
and
C
be sets of strings. Prove or disprove that
A
⋅
(
B

C
) =
A
⋅
B

A
⋅
C
.
Only one subset relation can be proved, namely we can prove that
A
⋅
B

A
⋅
C
⊆
A
⋅
(
B

C
).
It actually means that
A
⋅
B

A
⋅
C
is “smaller” then
A
⋅
(
B

C
), because there might be a
string that belongs to both
A
⋅
B
and
A
⋅
C
, although it can not be represented as some
prefix from
A
and a suffix that belong to both
B
and
C.
So, here is a counterexample that disproves
A
⋅
(
B

C
)
⊆
A
⋅
B

A
⋅
C
.
A
={
a
,
ab
},
B
={
b
},
C
={
λ
}.
B

C
={
b
},
A
⋅
(
B

C
)={
ab
,
abb
},
A
⋅
B
={
ab
,
abb
},
A
⋅
C
={
a
,
ab
}, and
A
⋅
B

A
⋅
C
= {
abb
}. So,
ab
∈
A
⋅
(
B

C
), but
ab
∉
A
⋅
B

A
⋅
C
, so it’s not the case,
that
always
A
⋅
(
B

C
)
⊆
A
⋅
B

A
⋅
C
.
2. Let
A
,
B
and
C
be sets of strings. Prove or disprove that if
A
⊆
B
, then
A
*
⊆
B
*
.
Assume
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 Spring '09
 Formal language, Regular expression, Kleene star

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