# I can't visualize what happens when we pump v and y in pumping lemma for $a^n b^n c^n$

If you need some context-: https://www.andrew.cmu.edu/user/ko/pdfs/lecture-11.pdf around page 7.

Case 1-: Say vxy contains ab

So when I pump v and y, what will get pumped? And how the result would be. I can understand the case for when vxy is all a's, all b's or all c's.

Case 2-: Say vxy contains bc

When I pump v and y, what happens? Help me visualize this.

My try-:

I will take case 1.

take i=2 then

We let

$$u=a^{n-k}$$

$$v=a^k$$

x=$$\in$$

y=$$b^l$$ k+l<=n

z=$$b^{n-l} c^n$$

Now I find $$u v^2 x y^2 z$$=? whwich gives $$a^{n+k} b^{n+l} c^n$$ which is obviously not in L. Am I right here?

Since $$vxy\in a^*b^*$$, and $$vy\neq \varepsilon$$, that means that $$v$$ and/or $$y$$ contains at least one $$a$$ or at least one $$b$$, and no $$c$$. That means that $$uxz$$ (pumping can also mean considering $$uv^0xy^0z$$) contains strictly more $$c$$'s than either $$a$$'s or $$b$$'s (or both), so $$uxz\notin \{a^nb^nc^n\mid n\in\mathbb{N}\}$$.