Why is {a^nb^n | n >= 0} not regular?

Question

In a CS course I\'m taking there is an example of a language that is not regular:

{a^nb^n | n >= 0}

I can understand that it is not regu

Answer 1

Assume L = {aⁿbⁿ | n ≥ 0} is regular. Then we can use the pumping lemma.

Let n be the pumping lemma number. Consider w = aⁿbⁿ∈L. The pumping lemma states that you can divide w into xyz such that xy ≤ n, y ≥ 1 and ∀ i∈ℕ₀: xyⁱz∈L.

Using the first two rules we can easily see that no matter how we divide w into xyz, y will always consist of only as and that it will contain at least one such letter. With rule 3 we conclude that a^n-kbⁿ∈L where k = |y| ≥ 1. But n-k ≠ n violates the definition of L, so that a^n-kbⁿ∉L. This is a contradiction