Question

In the Pumping Lemma, which condition must hold for the substring y in \( w = xyz \)?

• A. y must be at least the length of z
• B. y must be empty
• C. y is not empty
• D. y must contain only one type of symbol

Hint:

The middle part y is the piece you pump. Pumping is meaningless if that piece has length zero.

Answer 1

The Correct Option is C

Step 1: The pumping lemma for regular languages splits any long enough word as \( w = xyz \) with three conditions: \( |y| \ge 1 \), \( |xy| \le p \), and \( xy^iz \) is in the language for all \( i \ge 0 \).

Step 2: The middle part y is the piece that gets pumped (repeated or removed). If y were empty, pumping would do nothing, so the lemma requires \( |y| \ge 1 \), meaning y is not empty.

Step 3: Options about y being empty, longer than z, or single-symbol are not required conditions. The correct condition is that y is not empty, option (C).