# Dividing a String According to the Pumping Lemma

I have some questions about how a string can be divided into pieces according to the pumping lemma. I am learning from Michael Sipser’s book Introduction to the Theory of Computation, 3rd Edition. He states the pumping lemma as follows.

Pumping lemma If $$A$$ is a regular language, then there is a number $$p$$ (the pumping length) where if $$s$$ is any string in $$A$$ of length at least $$p$$, then $$s$$ may be divided into three pieces, $$s = xyz$$, satisfying the following conditions:

1. for each $$i≥0, xy^i z∈A$$,
2. $$|y|>0$$,
3. and $$|xy|≤p$$.

Sipser says that when $$s$$ is divided into $$xyz$$, either $$x$$ or $$z$$ may be $$ε$$, but not $$y$$ (condition 2).

He provides an example to show how the pumping lemma can be used to prove that the following language $$B$$ is not regular.

Let $$B=\{0^n 1^n | ≥ 0\}$$.

It is a proof by contradiction. He assumes $$B$$ is regular and presents three cases to show that it is not. In each case he splits a string $$s$$ from $$B$$ into $$xyz$$ in a different way. He chooses $$s$$ to be the string $$0^p1^p$$. In one case, Sipser lets $$y$$ consist only of 0s. Here are my questions:

1. In this case, is it possible for $$y$$ to consist of $$p$$ 0s and for $$x$$ to be the empty string if $$|xy|≤p$$ (condition 3)?
2. If so, do you agree that a case can be presented where $$s$$ is divided into $$xyz$$ such that $$x=ε,y=0^p,z=1^p$$?
3. If so, do you agree that the string $$xyyz$$ contains more 0s than 1s?
4. If so, do you agree that the string $$xyyz$$ is not in B?

I would greatly appreciate if you could answer all the questions in a numbered list format.