Solution

The correct answer is option 3) 1

Key Points

• The Pumping Lemma for regular languages helps us to prove whether a particular language is not regular. It provides a characteristic that all regular languages must satisfy. Here is the lemma:
• If a language L is regular, there exists a constant 'p' (the pumping length) such that for any string 's' in the language L, where the length of 's' (|s|) is greater or equal to 'p', 's' can be written as 'xyz' (s = xyz), satisfying these conditions:
  • For each 'i' ≥ 0, x(y^i)z is in L. This means that the string y can be "pumped" - repeated 'i' times for every nonnegative integer 'i' and the resulting string will still be in the language.
  • |y| > 0: The length of the string y must be at least 1. This ensures there are some characters to repeat when "pumping".
  • |xy| ≤ p: The combined length of strings x and y must be lesser or equal to the pumping length 'p'. This condition make sure we are only pumping within the first 'p' characters of 's' as we don't know what happens after 'p'.
• If a language fails to satisfy the Pumping Lemma conditions, it cannot be regular. Understanding why y's length has to be greater than 0 revolves around the concept that we are aiming to 'pump' or repeat the y section of the string. If y had no characters (|y| = 0), there would be nothing to repeat, which would make the pumping lemma concept pointless. Therefore, the minimum length of 'y' must be '1'.