   Chapter 12.2, Problem 54ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### 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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