A regular language over an alphabet ∑ is one that cannot be obtained from the basic languages using the operation

The statement seems to be incomplete. However, I can provide a general definition of a regular language.

A regular language over an alphabet ∑ is a type of formal language. It can be expressed using a regular expression, or equivalently, a deterministic or non-deterministic finite automaton.

The basic languages are the set of all strings over ∑ that contain a single symbol from ∑, and the language that contains the empty string.

The operations used to obtain regular languages from these basic languages are:

1. Union: If L1 and L2 are regular languages, then L1 ∪ L2 is also a regular language.
2. Concatenation: If L1 and L2 are regular languages, then L1.L2 is also a regular language.
3. Kleene Star: If L is a regular language, then L* (set of all strings that can be made by concatenating zero or more strings in L) is also a regular language.

So, a regular language over an alphabet ∑ is one that can be obtained from the basic languages using the operations of union, concatenation, and Kleene star.

The statement seems to be incomplete. However, I can provide some information based on what you've given.

A regular language over an alphabet ∑ is a language that can be expressed with a regular expression or a deterministic or non-deterministic finite automaton. It can be obtained from the basic languages using the operations of union, concatenation, and Kleene star (or repetition).

Here are the steps to form a regular language:

1. The empty set is a regular language.
2. For each symbol 'a' in the alphabet ∑, the singleton set {a} is a regular language.
3. If A and B are regular languages, then so are:
   • A union B (A ∪ B)
   • A concatenation B (AB)
   • A Kleene star (A*)
4. No other languages over ∑ are regular.

These operations allow us to build more complex regular languages from simpler ones.