Show that the language is regular

In summary, a regular language is a formal language that can be described by a regular expression or a finite automaton. To show that a language is regular, one can use methods like constructing a regular expression or a finite automaton, proving it can be generated by a regular grammar, or using the pumping lemma. The pumping lemma is a theorem that states a constant number exists for any regular language, allowing for the division of longer strings into smaller parts that can be repeated to create new strings in the language. A finite automaton is a theoretical model of computation while a regular expression is a notation for representing regular languages. While both can describe regular languages, they differ in structure and application. A language cannot be both regular and context-free, as
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

Let $$C_n=\{x \ \mid \ x \text{ is a binary number that is a multiple of } n\}$$

Show that for each $n \geq 1$, the language $C_n$ is regular. Could you give me some hints how we could show that??

Do we have to construct a NFA that accepts the language??
 
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  • #2
Let the set of states be $0,\dots,n-1$, i.e., possible remainders when a number is divided by $n$. Think about how the remainder changes when a number is read digit by digit right to left.
 

FAQ: Show that the language is regular

What is a regular language?

A regular language is a type of formal language that can be described by a regular expression or a finite automaton. It is a set of strings that can be generated by a regular grammar or recognized by a deterministic finite automaton.

How do you show that a language is regular?

To show that a language is regular, you can use various methods such as constructing a regular expression or a finite automaton that accepts the language, proving that the language can be generated by a regular grammar, or using the pumping lemma to show that the language is closed under concatenation.

What is the pumping lemma?

The pumping lemma is a theorem in formal language theory that states that for any regular language, there exists a constant number called the "pumping length" such that any string in the language longer than this length can be divided into smaller parts that can be repeated an arbitrary number of times to create new strings that are also in the language.

What is the difference between a finite automaton and a regular expression?

A finite automaton is a theoretical model of computation that consists of a set of states, transitions, and a start and accept state. It can be represented graphically as a directed graph. A regular expression, on the other hand, is a notation for representing regular languages using a combination of symbols and operators. While both can be used to describe regular languages, they are fundamentally different in their structure and application.

Can a language be regular and context-free at the same time?

No, a language cannot be both regular and context-free. Regular languages are a subset of context-free languages, meaning that all regular languages are also context-free. However, there are context-free languages that are not regular, such as the language of palindromes.

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