# 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 ) .

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

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Chapter 12.2, Problem 54ES
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