Describe the error in the following proofs that incorrectly conclude that = { | , ≥ } is not a regular language.
a. Proof (by contradiction). Assume that is a regular language. Let p be the pumping length for . Let be the string 0 p1 p . is a valid string because ∈ and || ≥ p. Consider all ways of dividing into x, y, and z. If y consists only of 0’s, then condition 3 of the pumping lemma does not hold because xy 2 z = 0 p+1 p ∉ (for > 0). If y contains one or more 1’s, then condition 2 of the pumping lemma does not hold because |xy| = |0 p1 | > p (for > 0). Therefore, the pumping lemma does not hold and is not a regular language.