# regular languages – Pumping lemma: why x in ∣xy∣ ≤ p?

Looking at the pumping lemma, I’ve noticed that in the string $$xy^pz$$, there seems to be no rule explicitly stated for $$x$$ and $$z$$. If I understand correctly, $$x$$ and $$y$$ are basically anything on the 2 sides of the string $$y^p$$ that we’re pumping and thus can be anything in $$L$$.

Rule 2 & 3 of the pumping lemma are:

• $$|y| geq 1$$
• $$|xy| leq p$$

Since $$|x| = 0$$ and $$|z| = 0$$ seem to be allowed, as they only need to be of non-negative length, we shouldn’t need $$x$$ in rule 2 and it can be rewritten as $$1 leq |y| leq p$$.

Are $$x$$ and $$y$$ not just a substitute for whatever are on the 2 sides of $$y^p$$ which we’re pumping? Why is $$x$$ in rule 2 if it doesn’t seem to make a difference? If $$x$$ is necessary, why is there no $$|yz| leq p$$?