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Discrete Mathematics With Applicat...

5th Edition
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ISBN: 9781337694193

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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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a. Let A be a finite-state automaton with input alphabet , and suppose L ( A ) is the language accepted by A. The complement of L ( A ) is the set of all strings over that are not in L ( A ) . Show that the complement of a regular language is regular by proving the following: L ( A ) is the language accepted by a finite-state automaton A, then there is a finite-state automaton A that accepts the complement of L ( A ) .

b. Show that the intersection of any two regular languages is regular as follows: First prove that if L ( A 1 ) and L ( A 2 ) are languages accepted by automata A 1 and A 2 , respectively, then there is an automaton A that accepts ( L ( A 1 ) ) c ( A ( A 2 ) ) c . Then use one of Dc Morgan’s laws for sets, the double complement law for sets, and the result of part (a) to prove that there is an automaton that accepts L ( A 1 ) L ( A 2 ) .

To determine

(a)

To prove:

That the complement of a regular language L(A) is also a regular language by showing that there exists a finite-state automaton accepts the complement of L(A).

Explanation

Given information:

A finite-state automaton (A) is given that accepts the language L(A) which is defined on an alphabet . The complement of L(A) is a set of strings on that not belong to L(A).

Formula used:

Kleene’s theorem for accepted language.

If any language is accepted by a finite-state automaton, then there exists a regular expression that defined the subjected language.

If any language can be expressed by a regular expression, then there exists a finite-state automaton that accepts the same language.

Proof:

Because L(A) is accepted by A, by the Kleene’s theorem there exists a regular expression that can be used to express L(A)

To determine

(b)

To prove:

That the intersection of two regular languages is regular by proving the union of complements of the two languages is accepted by a finite-state automaton.

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Chapter 12 Solutions

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Sect-12.1 P-11TYSect-12.1 P-12TYSect-12.1 P-1ESSect-12.1 P-2ESSect-12.1 P-3ESSect-12.1 P-4ESSect-12.1 P-5ESSect-12.1 P-6ESSect-12.1 P-7ESSect-12.1 P-8ESSect-12.1 P-9ESSect-12.1 P-10ESSect-12.1 P-11ESSect-12.1 P-12ESSect-12.1 P-13ESSect-12.1 P-14ESSect-12.1 P-15ESSect-12.1 P-16ESSect-12.1 P-17ESSect-12.1 P-18ESSect-12.1 P-19ESSect-12.1 P-20ESSect-12.1 P-21ESSect-12.1 P-22ESSect-12.1 P-23ESSect-12.1 P-24ESSect-12.1 P-25ESSect-12.1 P-26ESSect-12.1 P-27ESSect-12.1 P-28ESSect-12.1 P-29ESSect-12.1 P-30ESSect-12.1 P-31ESSect-12.1 P-32ESSect-12.1 P-33ESSect-12.1 P-34ESSect-12.1 P-35ESSect-12.1 P-36ESSect-12.1 P-37ESSect-12.1 P-38ESSect-12.1 P-39ESSect-12.1 P-40ESSect-12.1 P-41ESSect-12.2 P-1TYSect-12.2 P-2TYSect-12.2 P-3TYSect-12.2 P-4TYSect-12.2 P-5TYSect-12.2 P-6TYSect-12.2 P-7TYSect-12.2 P-8TYSect-12.2 P-9TYSect-12.2 P-10TYSect-12.2 P-1ESSect-12.2 P-2ESSect-12.2 P-3ESSect-12.2 P-4ESSect-12.2 P-5ESSect-12.2 P-6ESSect-12.2 P-7ESSect-12.2 P-8ESSect-12.2 P-9ESSect-12.2 P-10ESSect-12.2 P-11ESSect-12.2 P-12ESSect-12.2 P-13ESSect-12.2 P-14ESSect-12.2 P-15ESSect-12.2 P-16ESSect-12.2 P-17ESSect-12.2 P-18ESSect-12.2 P-19ESSect-12.2 P-20ESSect-12.2 P-21ESSect-12.2 P-22ESSect-12.2 P-23ESSect-12.2 P-24ESSect-12.2 P-25ESSect-12.2 P-26ESSect-12.2 P-27ESSect-12.2 P-28ESSect-12.2 P-29ESSect-12.2 P-30ESSect-12.2 P-31ESSect-12.2 P-32ESSect-12.2 P-33ESSect-12.2 P-34ESSect-12.2 P-35ESSect-12.2 P-36ESSect-12.2 P-37ESSect-12.2 P-38ESSect-12.2 P-39ESSect-12.2 P-40ESSect-12.2 P-41ESSect-12.2 P-42ESSect-12.2 P-43ESSect-12.2 P-44ESSect-12.2 P-45ESSect-12.2 P-46ESSect-12.2 P-47ESSect-12.2 P-48ESSect-12.2 P-49ESSect-12.2 P-50ESSect-12.2 P-51ESSect-12.2 P-52ESSect-12.2 P-53ESSect-12.2 P-54ESSect-12.3 P-1TYSect-12.3 P-2TYSect-12.3 P-3TYSect-12.3 P-4TYSect-12.3 P-5TYSect-12.3 P-1ESSect-12.3 P-2ESSect-12.3 P-3ESSect-12.3 P-4ESSect-12.3 P-5ESSect-12.3 P-6ESSect-12.3 P-7ESSect-12.3 P-8ESSect-12.3 P-9ESSect-12.3 P-10ESSect-12.3 P-11ESSect-12.3 P-12ESSect-12.3 P-13ESSect-12.3 P-14ESSect-12.3 P-15ESSect-12.3 P-16ESSect-12.3 P-17ESSect-12.3 P-18ES