7: Properties of regular languages
Theorem 1
The set of regular languages are closed
under
1. Concatenation (
L
1
and
L
2
regular, then so is
L
1
L
2
),
2. Union (
L
1
and
L
2
regular, then so is
L
1
∪
L
2
),
3. Kleene star (
L
regular, then so is
L
∗
),
4. Complementation (
L
regular, then so is
L
=Σ
∗
−
L
), and
5. Intersection (
L
1
and
L
2
regular, then so is
L
1
∩
L
2
).
1
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1.3. We have seen two ways of proving these statements
in earlier lecture notes.
The Frst way is to show
that there exists a regular expression that represents
the resulting language, the second way is to show
that there exists an (N)±A that accepts the resulting
language.
4. Since
L
is regular,
L
is accepted by some D±A
M
.
Let
M
±
be the same as
M
except that:
A state is a Fnal state in
M
±
i² it is not a
Fnal state in
M
.
L
is accepted by the D±A
M
±
. Hence,
L
is regular.
5.
L
1
∩
L
2
=
L
1
∪
L
2
.
Since the set of all regular languages is closed under
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 Spring '10
 Prof.Tai
 Formal language, Regular expression, Regular language, Nondeterministic finite state machine, decimal representation, Kleene star

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