   Chapter 12.1, Problem 6TY ### 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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# The function that associates a language to each regular expression over an alphabet ∑ is defined recursively. The base for the definition is the statement that L ( ∅ ) = _______ , L ( λ ) = _______ , and L ( a ) = _______ for every a in ∑ . The recursion for the definition specifies that if L ( r ) and L ( r ' ) are the languages defined by the regular expressions r and r’ over ∑ , then L ( r r ' ) = _______ , L ( r | r ' ) = _______ , and L ( r * ) = _______ .

To determine

To fill in the blanks of the statement on languages and regular expression.

The function that associated a language to each regular expression over an alphabet Σ is defined recursively. The base for the definition is the statement that L()=____,L(λ)=_____, and L(a)=____ for every a in Σ. The recursion for the definition specifies that if L(r) and L(r') are languages defined by the regular expressions r and r' over Σ, then L(rr')=___,L(r|r')=___ and L(r*)=___ ”.

Explanation

Given information:

The language L and regular expressions r and r' are defined on an alphabet Σ.

By the notation L(), we denote language that is formed by a set of string defined on a particular alphabet Σ. A language is always a set of strings.

Also, the notation is used to denote the null set.

Then, a language defined by null set has only the strings formed by null set. Hence it has no strings actually, Therefore, the language defined by a null set is also a null set.

L()=____

Also, the notation λ is used to denote a string with zero length, (a string has no characters). Hence, the language defined by λ contains set of strings of zero length.

Hence, L(λ)=____{λ}

When a is a string defined on alphabet Σ, the language defined by a has only one string a.

L(a)=____{a}

Until r is a regular expression defined on Σ, rr' denotes the concatenation of two regular expressions. Hence, L(rr') is the language of that concatenated two regular expressions. Therefore, L(rr') concatenates the languages defined by r and r'

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