Chapter 12.2, Problem 54ES

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Textbook Problem

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a. Let *A *be a finite-state automaton with input alphabet
*A. *The complement of
*. *Show that the complement of a regular language is regular by proving the following:
*A, *then there is a finite-state automaton
*. *

b. Show that the intersection of any two regular languages is regular as follows: First prove that if
*, *respectively, then there is an automaton *A *that accepts
*. *Then use one of Dc Morganâ€™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

To determine

**(a)**

**To prove:**

That the complement of a regular language

Explanation

**Given information:**

A finite-state automaton

**Formula used:**

Kleeneâ€™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

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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