What are the closure properties of regular languages?

The closure properties of regular languages refer to the properties that are preserved under certain operations on regular languages. These closure properties are important in formal language theory as they allow us to manipulate and combine regular languages while ensuring that the resulting language remains regular.

The closure properties of regular languages include:

1. Union: The union of two regular languages is also a regular language. In other words, if L1 and L2 are regular languages, then their union L1 ∪ L2 is also a regular language.

2. Concatenation: The concatenation of two regular languages is also a regular language. If L1 and L2 are regular languages, then their concatenation L1L2 is also a regular language.

3. Kleene Star: The Kleene star operation on a regular language L, denoted as L*, generates a new regular language that includes all possible concatenations of zero or more strings from L. Therefore, if L is a regular language, then L* is also a regular language.

4. Intersection: The intersection of two regular languages is also a regular language. If L1 and L2 are regular languages, then their intersection L1 ∩ L2 is also a regular language.

5. Complementation: The complement of a regular language L, denoted as L', is the set of all strings that are not in L. If L is a regular language, then its complement L' is also a regular language.

These closure properties demonstrate that regular languages are closed under various operations, allowing us to perform operations on regular languages and still obtain regular languages as a result.