A regular language over an alphabet ∑ is closed under following operation(s):

Solution

Regular languages are a fundamental class of formal languages that are closed under several key operations. This means that if you apply these operations to regular languages, the resulting language remains regular.

1. Union

If L1 and L2 are regular languages, then their union (L1 ∪ L2) is also a regular language. This can be proven using finite automata by constructing a new machine that accepts strings from either L1 or L2.

2. Concatenation

The concatenation of two regular languages (L1 · L2) is also regular. This operation involves forming all possible strings by taking a string from L1 and appending a string from L2. Finite automata can be designed to handle this sequence.

3. Kleene Star

The Kleene star operation (L*) on a regular language L produces a regular language. This operation allows for zero or more repetitions of strings from L. It is a core property used in defining regular expressions.

Since regular languages are closed under union, concatenation, and the Kleene star operation individually, the correct answer is that they are closed under all of the mentioned operations.