Question

Suppose that B is a regular language over an alphabet Σ. Let

C = { ∈ ∑* a ∈ ∑* such that aw Є }
(a) Prove that we may choose a such that |a| < p in the definition of C, where p is the pumping length of B. That is, prove that we have C = { ∈ ∑* ∣ ∃a ∈ ∑* such that aw B and ∣a∣

Answer 1

The Pumping Lemma for regular languages proves that for a regular language B and a set C defined by strings that can be appended to some string a resulting in a string in B, one can choose a with length less than the pumping length p of B.

Step-by-step explanation:

The question pertains to formal languages and automata theory, specifically to the Pumping Lemma for regular languages. Given an alphabet Σ and a regular language B over Σ with pumping length p, the set C consists of strings that can be appended to some string a such that the concatenation is in B. To prove that a can be chosen with length strictly less than p, we can use the Pumping Lemma which states that for any string s in a regular language with length at least p, we can split s into three parts x, y, and z such that xynz is in the language for any n ≥ 0, and |xy| ≤ p with |y| > 0. If aw is in B and |aw| ≥ p, we can pump y of aw and show that any a can be chosen with length less than p because we are only working with the prefix of length p or less, which implies |a| < p.