Question

Use pumping lemma to prove that the language {aⁿb²ⁿ| n>0, and w is in {a, b}^* } is not a regular language.

Answer 1

Given Language - {aⁿb²ⁿ| n>0, and w is in {a, b}^* }

Pumping Lemma states that

For any language L, there exists an integer n, such that for all x ∈ L with |x| ≥ n, there exists u,v, w ∈ Σ*, such that x = uvw, and
(1) |uv| ≤ n
(2) |v| ≥ 1
(3) for all i ≥ 0: uv^ⁱw ∈ L

To prove that the language is not regular we have show that there exist alteast one string such that it is not in the language (i.e it has to fail pumping lemma Test)

let X=ab² (n=1) where pumping length = 3

u= ,v = a and w = b²

and on pumping 'a'(i.e v) once, we get a²b² which doesn't belong to the given language

so the given is not regular language