Pumping Lemma and Fragmenting the String

The Pumping Lemma is a powerful tool used in formal language theory to prove that certain languages are not regular. It provides a way to show that a language does not satisfy the conditions of regularity by considering the possible divisions of a string into parts.

The Steps:

To apply the Pumping Lemma, we need to follow these steps:

1. Assume L is a regular language.
2. Select a "pumping length" p for L.
3. Choose a string w that belongs to L and has a length of at least p.
4. Divide the string w into parts: uvxyz.
5. Consider the possible cases for the division of the string w:
- The string can be divided into 3 parts: uvxyz.
- The string can be divided into 5 parts: uvwxy.
- The string can be divided into 2 parts: xy.
- The string can be divided into 6 parts: uvwxyz.
- And so on...

Fragmenting the String:

In this specific question, we are considering a string w that belongs to the language L. The question asks us to fragment the string into a specific number of parts. The correct answer is option 'C', which means we need to fragment the string into 3 parts.

By fragmenting the string into 3 parts, we can analyze and manipulate the substrings u, v, x, and y to demonstrate that the language L does not satisfy the conditions of regularity. This is the essence of the Pumping Lemma, where we can show that no matter how we manipulate the substrings, we will eventually reach a point where the resulting string is not in the language L.

Therefore, by considering a string w that belongs to L and fragmenting it into 3 parts, we can effectively apply the Pumping Lemma to prove that the language L is not regular.